Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Can $10101\dots1$ be a perfect square in any base? Inpired by another question and it's answer I started to wonder if it's true in other bases as well. I've at least not found any base $b$ where $10101\dots1 = (b^{2k}-1)/(b^2-1)$ (where $k>1$) is a perfect square anytime.
From the answers we can conclude that $8|b^2$ w... | Just some ideas, too long for comments.
Summary: No such square exists if either $b$ or $k$ is even.
This can be written as $(b^k)^2-(b^2-1)y^2=1$.
The set of positive integer solutions to the Pell equation $x^2-(b^2-1)y^2=1$ are exactly the values:
$$x+y\sqrt{b^2-1}=\left(b+\sqrt{b^2-1}\right)^n; n\in\mathbb Z^{+}$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2421547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 1,
"answer_id": 0
} |
Prove $\lim_{x\to 2} \sqrt{x^2+5} = 3$ As stated in the title, I need to prove that $\lim_{x\to 2} \sqrt{x^2+5} = 3$ using only the precise definition of a limit.
For any given $\varepsilon \gt 0$, there exists a $\delta = $
Such that $0 \lt \lvert x-2 \rvert \lt \delta \Rightarrow \lvert \sqrt{x^2+5} \rvert \lt \vare... | We should prove that,for any given $ε>0$, there exists a $δ$ if $0 \lt \lvert x-2 \rvert \lt \delta \Rightarrow \lvert \sqrt{x^2+5} -3\rvert \lt \varepsilon$
$$\left| \sqrt { { x }^{ 2 }+5 } -3 \right| =\left| \frac { { x }^{ 2 }-4 }{ \sqrt { { x }^{ 2 }+5 } +3 } \right| =\left| \frac { { \left( x-2 \right) }\left( x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2423938",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Evaluate $\int \frac{dx}{x(x-1)^3(x-2)^2}$ Evaluate $$I=\int \frac{dx}{x(x-1)^3(x-2)^2}$$ without using tedious partial fractions.
My Try:
we have $$1=\left((x-1)-(x-2)\right)^2$$ So
$$=\int \frac{\left((x-1)-(x-2)\right)^2dx}{x(x-1)^3(x-2)^2}$$ So
$$I=\int \frac{dx}{x(x-1)(x-2)^2}+\int \frac{dx}{x(x-1)^3}-2\int \fra... | You can reduce the labour a little by writing
$\dfrac{1}{x(x-1)^3(x-2)^2} = \dfrac{(1-x+x^2)+x(x-1)}{x(x-1)^3(x-2)^2}$
$=\dfrac{x(x-2)^2-(x-1)^3}{x(x-1)^3(x-2)^2} + \dfrac{1}{(x-1)^2(x-2)^2}$
$=\dfrac{1}{(x-1)^3} - \dfrac{1}{x(x-2)^2}+\dfrac{1}{(x-1)^2(x-2)^2}$
$=\dfrac{1}{(x-1)^3}- \dfrac{1}{x(x-2)^2}+ \dfrac{1}{(x-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2424265",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How many 6-digit numbers have distinct digits but no consecutive digits both odd? How many 6-digit natural numbers exist with the
distinct digits and two arbitrary consecutive digits can not be simultaneously odd numbers?
I have tried to set up recurrence relation , by considering a valid 5 digit number satisfying the ... | One way to do it is by looking at cases, as @lulu suggested. There are $3$ basic cases: $1$ odd number, $2$ odd numbers or $3$ odd numbers. I'll go through each case below, splitting each case into scenarios starting with an odd number (O) and scenarios starting with an even number (E). The reason for this split is tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2425493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding derivative of $\frac{x}{x^2+1}$ using only the definition of derivative I think the title is quite self-explanatory. I'm only allowed to use the definition of a derivative to differentiate the above function. Sorry for the formatting though.
Let $f(x) = \frac{x}{x^2+1}$
$$
\begin{align}
f'(x)&= \lim_{h\to 0} \... | \begin{eqnarray}
{1 \over h } ({x+h \over 1 + (x+h)^2} - {x \over 1+x^2}) &=&
{1 \over h } { (x+h)(1+x^2) - (1+(x+h)^2)x\over (x+x^2) (1 + (x+h)^2} \\
&=& {1 \over h } { h -h x^2 - h^2 x\over (x+x^2) (1 + (x+h)^2} \\
&=& { 1 - x^2 - h x\over (x+x^2) (1 + (x+h)^2}
\end{eqnarray}
The limit follows by taking $ h \to 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2426142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
find the total number of possible way to reach to a particular sum suppose you have given a sum like : 5.
we have to find the total number of possible way to reach to 5.
for example
1 + 1 + 1 + 1 + 1 = 5
2 + 1 + 1 + 1 = 5
1 + 2 + 1 + 1 = 5
1 + 1 + 2 + 1 = 5
1 + 1 + 1 + 2 = 5
2 + 2 + 1 = 5
2 + 1 ... | (Related to stars and bars, but a bit more explicit).
Represent $5$ as $u\;u\;u\;u\;u$.
Between any two $u$s you can insert or not insert a separator. Once you have done this you will have uniquely determined an ordered sum to $5$.
For instance $u*u\;u\;u*u$ corresponds to $1+3+1$.
Or conversely, $2+3$ corresponds ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2426624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Solve the following nonlinear system of equations:$ xy+xz+yz=12 , xyz=2+x+y+z$
Solve the following system of equations in $\Bbb R^+$:
$$
\left\{
\begin{array}{l}
xy+yz+xz=12 \\
xyz=2+x+y+z\\
\end{array}
\right.
$$
I did as follows.
First equation yields to: $(x+y+z)^2-x^2-y^2-z^2=24$. Now, using second equat... | The condition gives $$xyz=2+x+y+z$$ or
$$xyz=2+\frac{(x+y+z)(xy+xz+yz)}{12}$$ or
$$12xyz=24+\sum_{cyc}(x^2y+x^2z+xyz)$$ or
$$3xyz=24+\sum_{cyc}(x^2y+x^2z-2xyz),$$
which gives
$$3xyz-24=\sum_{cyc}(x^2y+x^2z-2xyz)=\sum_{cyc}z(x-y)^2\geq0.$$
Thus, $xyz\geq8$ and the equality occurs for $x=y=z$.
On the other hand, by AM-GM... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2427470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 1
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For what values of $a$ does the system have infinite solutions? Find the solutions. The system is
$$\left\{
\begin{array}{rcr}
x+ay+z & = & 1 \\
ax+y+z & = & 1+a \\
x-y+z & = & 2+a
\end{array}
\right.$$
After row reducing I got
$$\left\{
\begin{array}{rcr}
x+ay+z & = & 1 \\
-(1+a)y+0 & = & 1+a \\
(1-a)z& = & 2-a + (1... | It's simpler to perform the full row reduction for the augmented matrix. We'll put the third equation at top:
\begin{gather}
\begin{bmatrix}
1&-1&1&|&2+a\\1&a&1&|&1\\
a&1&1&|&1+a\end{bmatrix}\rightsquigarrow
\begin{bmatrix}
1&-1&1&|&2+a\\0&1+a&0&|&-1-a\\
0&1+a&1-a&|&1-a-a^2\end{bmatrix} \\[1ex]
\rightsquigarrow \begin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2427722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Find the number of permutations of the word MATHEMATICS that satisfy at least one of three restrictions Find the number of permutations of letters of the word ’MATHEMATICS’ where: both letters T
are before both letters A or both letters A are before both letters M or both letters M are before
letter E.
We have 2 Ms 2 A... | Your strategy is correct but not all of your numbers are.
I will use the same notation that you did.
Let $A$ be the event that both Ts appear before both As.
Let $B$ be the event that both Ms appear before both As.
Let $C$ be the event that both Ms appear before E.
Then, as you observed, the number of permutatio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2428464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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In Epsilon-Delta proof, why is $3x^2|y|/(x^2 + y^2) \Leftarrow 3|y| = 3\sqrt{y^2} \Leftarrow 3\sqrt{x^2 + y^2} $ This is part of a Epsilon-Delta proof to show that: $$\lim_{(x, y) \to 0}\frac{3x^2y}{x^2 + y^2} = 0$$ The proof began with: $$\frac {3x^2|y|}{(x^2 + y^2)} < \epsilon$$ It was then pointed out that $x^2 \Lef... | Assume $x,y\in\mathbb{R}$. First, you can start saying that $x^{2}\geq 0$, then adding $y^{2}$ in both sides of the inequality you get:
$$ x^{2}+y^{2}\geq y^{2} \Rightarrow \sqrt{x^{2}+y^{2}}\geq\sqrt{y^{2}}$$
because $f(t)=\sqrt{t}$ is monotone increasing on its domain. Now use that $g(t)=3t$ is also monotone increasi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2428595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that if the columns of the $m\times n$ matrix $A$ are linearly independent, then $Ax=b$ has at most one solution.
Prove that if the columns of the $m\times n$ matrix $A$ are linearly independent, then $Ax=b$ has at most one solution.
I was thinking that if the $n$ columns are linearly independent, then the dime... | Let $A = \begin{bmatrix}\mathbf{a}_1 & \mathbf{a}_2 & \cdots & \mathbf{a}_n\end{bmatrix}$ and $\mathbf{x} = \begin{bmatrix}x_1\\x_2\\\vdots\\x_n\end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix}b_1\\b_2\\\vdots\\b_m\end{bmatrix}$.
Now, $A\mathbf{x} = x_1 \mathbf{a}_1 + x_2 \mathbf{a}_2 + \cdots + x_n\mathbf{a}_n$.
Let $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2428981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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How to test for convergence for the harmonic series with irregular (binomial) sign changes? How would you test for convergence with a series such as this using the alternating series test?
$1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\frac{1}{7}...-\frac{1}{10}+\frac{1}{11}...+\frac{1}{15}-....$
I hav... | I hope I'm interpreting this correctly: you have one $+$ term, then two $-$, three $+$, four $-$, etc. The start and end of a run of $2k+1$ $+$ terms are $1/(2 k^2 + k + 1)$ and $1/(2 k^2 + 3 k + 1)$ and the start and end of the next run of $2k+2$ $-1$ terms are $-1/(2 k^2 + 3 k + 2)$ and $-1/(2 k^2 + 5 k + 3$.
Thus t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2430060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A sequence divisible by 9 I was trying to solve this series by mathematical induction for every $n$ from $\Bbb N$ : $u_n=n4^{n+1}-(n+1)4^n+1$ is divisible by $9$.
The initiation was pretty easy, but I only managed to prove $u_{n+1}=3k$ while $k$ is an integer and I don't think if it's divisible by $3$ implies that it... | Base case: $1\cdot 4^2 - 2\cdot 4^1 + 1 = 9$ is divisible by $9$.
Induction step: Assume it is true for $n = a$, say $u_a = 9k$. Now to see what we get for $n = a+1$:
$$
u_{a+1} = (a+1)4^{a+2} - (a+2)4^{a+1} + 1\\
= a4^{a+2} + 4^{a+2} - (a+1)4^{a+1} - 4^{a+1} + 1\\
= 4(a 4^{a+1} - (a+1)4^a + 1) + 4^{a+2} - 4^{a-1} - 3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2430397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 2
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Integrating a Rational Function with a Square Root : $ \int \frac{dx}{\sqrt{x^2-3x-10}} $
Integrate $$ \int \frac{dx}{\sqrt{x^2-3x-10}}. $$
I started off with the substitution $\sqrt{x^2-3x-10} = (x-5)t$. To which I got
$$ x = \frac{2+5t^2}{t^2-1} \implies \sqrt{x^2-3x-10} = \left(\frac{2+5t^2}{t^2-1} - 5\right)t = \... | This is an alternative method to compute the given integral.
$$I =\int{\frac{dx}{\sqrt{x^2-3x-10}}} = \int{\frac{dx}{\sqrt{(x-3/2)^2-(7/2)^2}}}\\=\log\left|(x-3/2)+\sqrt{(x-3/2)^2-(7/2)^2}\right|+c.$$
Here $$\int{\frac{dy}{\sqrt{y^2-a^2}}}=\log\left|y+\sqrt{y^2-a^2}\right|+c.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2431066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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$a,b,c$ are positive reals and distinct with $a^2+b^2 -ab=c^2$. Prove $(a-c)(b-c)<0$
$a,b,c$ are positive reals and distinct with $a^2+b^2 -ab=c^2$.
Show that $(a-c)(b-c)<0$.
This is a question presented in the "Olimpiadas do Ceará 1987" a math contest held in Brazil. Sorry if this a duplicate.
Given the assumptions,... | we get $$c=\sqrt{a^2+b^2-ab}$$ since $c>0$ and $$a^2+b^2>ab$$ from here we get
the product
$$(a-\sqrt{a^2+b^2-ab})(b-\sqrt{a^2+b^2-ab})<0$$
we have two cases.
1) $$a>\sqrt{a^2+b^2-ab}$$ and $$b<\sqrt{a^2+b^2-ab}$$ after squaring we get $$a>b$$
2) $$a<\sqrt{a^2+b^2-ab}$$ and $$b>\sqrt{a^2+b^2-ab}$$
after squaring we get... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2431928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 4
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Algebraic solution of complex equation For solving algebraically any complex equation involves two components for the real & imaginary parts. Let the real part be - $a$, imaginary part - $b$.
For the complex equation $$x^3 = 1-i $$
Substituting $x = a +bi$, we get: $$(a+bi)^3 = 1 - i.$$
Expanding the l.h.s. :
$$a^3 -(... | Hint:
$$a^3-3b^2a=b^3-3a^2b=1 \implies a^3-b^3+3a^2b-3b^2a=0$$ which implies
$$(a-b)(a^2+ab+b^2)+3ab(a-b)=0 \implies (a-b)(a^2+4ab+b^2)=0.$$
As such, $a=b$ or $a^2 + 4ab+b^2 = a^2+4ab+4b^2-3b^2 = 0 \implies a = (-2 \pm \sqrt{3})b.$
If $a=b$ then $a^3-3b^2a = -2a^3=1 \implies \color{red}{a=b=-2^{-1/3}}$.
I leave the ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2435350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 1
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Help clarifying the steps to find the derivative of $y=(3x+1)^3(2x+5)^{-4}$ For the problem $y=(3x+1)^3(2x+5)^{-4}$ do I use the chain, quotient and product rules? If so how do I know what parts to break up and where the rules apply? For instance would I consider $f(x)$ to be $(3x+1)^3$ and $g(x)$ to be $(2x+5)^{-4}$... | Hint: You are free to apply either the quotient rule or the product rule, both of them are convenient to use.
Product rule: Set
$$f(x)=(3x+1)^3\qquad\text{and}\qquad
g(x)=(2x+5)^{-4}
$$ and consider
\begin{align*}
\left(f(x)\cdot g(x)\right)^\prime=\left((3x+1)^3\cdot(2x+5)^{-4}\right)^\prime
\end{align*}
Quotien... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2436600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Proof by induction (Combinatorics) This is my first question on here so please bear with me, thank you.
Prove that for all positive integers $n$
$$2(1+2+...+n)^4 = (1^5 + 2^5 +...+ n^5) + (1^7 + 2^7 +...+ n^7)$$
After establishing the base case, I proceeded to the induction:
$2(1+2+...+(n+1))^4 = (1^5 + 2^5 +...+ (... | You must prove $$2\left(\sum_{i=1}^{n+1}i\right)^4 = \sum_{i=1}^{n+1}i^5 + \sum_{i=1}^{n+1}i^7.$$
Let $S_n = \sum_{i=1}^{n}i = \frac{n(n+1)}{2}$
You have $$2\left(\sum_{i=1}^{n+1}i\right)^4 = 2\left(\left(\sum_{i=1}^{n}i\right)^4 + 4(n+1)S_n^3 + 6(n+1)^2S_N^2 + 4(n+1)^3S_n + (n+1)^4 \right).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2436813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Calculate the line integral $I(a,b)=\int_{x^2+y^2=R^2}\limits\ln\frac{1}{\sqrt{(x-a)^2+(y-b)^2}} ds\quad(a^2+b^2\ne R^2).$ Calculate the line integral $$I(a,b)=\int_{x^2+y^2=R^2}\limits\ln\frac{1}{\sqrt{(x-a)^2+(y-b)^2}} ds\quad(a^2+b^2\ne R^2).$$
The parametrized integral path can be given as
$$x=R\cos t,y=R\sin t,t\i... | As noted in @Dattier answer,
\begin{align}
I&=-\frac{1}{2}\int_{0}^{2\pi}R\ln(a^2+b^2+R^2-2aR\cos t-2bR\sin t) dt\\
&=-\frac{1}{2}\int_{0}^{2\pi}R\ln(a^2+b^2+R^2-2R\sqrt{a^2+b^2}\cos t) dt\\
\end{align}
If $a^2+b^2<R^2$,
$$ \ln(a^2+b^2+R^2-2R\sqrt{a^2+b^2}\cos t)=2\ln R +\ln(1-2z\cos t +z^2)$$
with $z=\frac{\sqrt{a^2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2437653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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For each $a \in \Bbb{Z}$ work out $\gcd(3^{16} \cdot 2a + 10, 3^{17} \cdot a + 66)$ The problem is the following: For each $a \in \Bbb{Z}$ work out $\gcd(3^{16} \cdot 2a + 10, 3^{17} \cdot a + 66)$
This is what I have at the moment:
Let's call $d = \gcd(3^{16} \cdot 2a + 10, 3^{17} \cdot a + 66)$
Then $d \ \vert \ 3^{1... | Hint $\ \ \ (d,102) = (d,\,2\cdot 3\cdot 17) = (d,2)\,(d,3)\,(d,17).\, $ Computing these gcds by Euclid & Fermat
$\qquad\quad\ \begin{align} (d,2) &= (\color{#c00}{3^{16}(2a)\! +\! 10}, \ \color{#0a0}{3^{17} a\! +\! 66},\,2) = (\color{#c00}0,\color{#0a0}a,2) = (a,2)\\[.2em]
(d,3) &= (\color{#c00}1,\ \color{#0a0}0,\ 3) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2437747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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Solve nonlinear equation $-xu_x+uu_y=y$ Solve nonlinear equation
$$\left\{\begin{matrix}
-xu_x+uu_y=y & \\
u(x,2x)=0&
\end{matrix}\right.$$
using method of characteristic curves
my attempt:
for the pde we can write it as
$\frac{dx}{-x}=\frac{dy}{u}=\frac{du}{y}$
but i cant solve little please..thank you
| The given partial differential equation is:
$$\begin{matrix}
-xu_x+uu_y=y & \\
u(x,2x)=0.&
\end{matrix}$$
Finding the general solution
Using the method of characteristics we obtain:
$$\dfrac{dx}{-x}=\dfrac{dy}{u}=\dfrac{du}{y} \qquad (1)$$
We solve the right part of the previous equality:
$$\dfrac{dy}{u}=\dfrac{du}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2443958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Complex quintic equation Given the equation $x^5=i$, I need to show by both algebraic and trigonometrical approaches that
$$\cos18^{\circ}=\frac{\sqrt{5+2\sqrt5}}{\sqrt[5]{176+80\sqrt5}}$$
$$\sin18^{\circ}=\dfrac1{\sqrt[5]{176+80\sqrt5}}$$
Trying by trigonometric approach,
$x^5$ = i $\;\;\;\;$ -- eqn. (a)
=> x = $... | Let's start with equation $x^{5}-1=0$ whose one root is $x=\cos(2\pi/5)+i\sin(2\pi/5)$. The equation can be written as $$(x-1)(x^{4}+x^{3}+x^{2}+x+1)=0$$ The first factor gives the root $x=1$ and the second factor leads to the equation $$x^{2}+x^{-2}+x+x^{-1}+1=0$$ Putting $y=x+x^{-1}$ we get $$y^{2}+y-1=0$$ or $$y=\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2445926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
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Seeking the maximal parameter value s.t. two-variable inequality still holds Consider the expression
$$\frac 12\left(\frac{a^2}b+\frac{b^2}a\right)$$
in two variables $\,a,b\,$ residing in $\,\mathbb R^{>0}$.
The arithmetic mean $\,\frac{a+b}2\,$ is a lower bound for it
$\big[$ one has $\,a^2b\le(2a^3+b^3)/3\,$ by AM-G... | The following inequality is true.
Let $a$ and $b$ be positive numbers.
Prove that
$$\frac{1}{2}\left(\frac{a^2}{b}+\frac{b^2}{a}\right)\geq\sqrt[9]{\frac{a^9+b^9}{2}}.$$
Indeed, we need to prove that
$$(a^3+b^3)^9\geq256(a^9+b^9)a^9b^9$$ or
$$(a^3+b^3)^8\geq256(a^6-a^3b^3+b^6)a^9b^9.$$
Now, let $a^6+b^6=2ua^3b^3.$
He... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2447859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
What will be the complete area of curve $a^2x^2=y^3(2a-y)$ I want to calculate the area of $a^2x^2=y^3(2a-y)$ using double integral.
I decided to guess the curve and wrote the equation is:-
$$\pm x=\frac{y\sqrt{y(2a-y)}}{a}$$
which means the curve is symmetric about $y$ axis and $x=0$ at $y=0,2a$.
Thus, area should be... | Try to use $$\int (f(x))^a f^{(1)}(x) = \frac{f(x)^{a+1}}{(a+1)}$$
and
$$\int \sqrt{a^2-x^2}=\frac{a^2}{2} \arcsin(x/2) + \frac{x}{2}\sqrt{a^2-x^2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2448874",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How to evaluate the sum : $\sum_{k=1}^{n} \frac{k}{k^4+1/4}$ I have been trying to figure out how to evaluate the following sum:
$$S_n=\sum_{k=1}^{n} \frac{k}{k^4+1/4}$$
In the problem, the value of $S_{10}$ was given as $\frac{220}{221}$.
I have tried partial decomposition, no where I go. Series only seems like it t... | Try to break the denominator into product of two factors:
$$\begin{align}
4k^4 + 1 &= (2k^2)^2 + 1 + 2 (2k^2) - 2 (2k^2) \\
&= (2k^2 +1)^2 - (2k)^2 \\
&= (2k^2 +2k +1)(2k^2 -2k+1)
\end{align}$$
Using this we see the general term as:
$$T_k = \dfrac{1}{2k^2-2k+1} - \dfrac{1}{2k^2+2k+1} \\
T_{k+1} = \dfrac{1}{2k^2+2k+1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2449134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 0
} |
How to compute symmetrical determinant I'm learning of determinants and am trying to find a trick to compute this one
\begin{pmatrix}
2 & 1 & 1 & 1 & 1\\
1 & 3 & 1 & 1 & 1\\
1 & 1 & 4 & 1 & 1\\
1 & 1 & 1 & 5 & 1\\
1 & 1 & 1 & 1 & 6
\end{pmatrix}
I expanded it out and got $349$ but I feel there must b... | Your matrix is in a very nice form. As an alternate answer, you can use the matrix determinant lemma in this case to do the calculation quite cleanly.
Let $\mathbf{M}$ be the matrix you provided. Then $\mathbf{M}$ = $\mathbf{A} + \mathbf{j}\mathbf{j^t}$,
where
$$
\mathbf{A} =
\begin{pmatrix}
1 & 0 & 0 & 0 & 0\\
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2449842",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Proof that $\binom{n + 3}{4} = n + 3 \binom{n + 2}{4} - 3 \binom{n + 1}{4} + \binom{n}{4}$. I was trying to count the number of equilateral triangles with vertices in an regular triangular array of points with n rows. After putting the first few rows into OEIS, I saw that this was described by A000332: $\binom{n}{4} = ... | The identity is equivalent to
\begin{align*}
n&=\binom{3}{0}\binom{n + 3}{n-1}-\binom{3}{1} \binom{n + 2}{n-2} + \binom{3}{2}\binom{n + 1}{n-3} - \binom{3}{3}\binom{n}{n-4}\\
&=\sum_{k=0}^3 (-1)^k\binom{3}{k}\binom{(n-k)+3}{(n-k)-1}=[x^n](1-x)^3\cdot \frac{x}{(1-x)^5}=[x^{n+1}]\frac{1}{(1-x)^2}
\end{align*}
which holds... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2451199",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 0
} |
How to calculate the nth term and the sum in this series if the common difference between them isn't explicit? I have this series comprised of
$$1,2,5,10,17,26,...$$,
and so on. So far i have found that they add up in intervals being odd numbers. But I don't know how to find, let's say the 16th term and the sum up to t... | As Guarang Tandon says. You can create a table of differences between consecutive terms and then repeat that process on the list you just created.
\begin{array}{|l|c|ccccccc|}
\hline
\text{index} & n & 0 & 1 & 2 & 3 & 4 & 5 & \dots\\
\hline
\text{sequence} & f_n & 1 & 2 & 5 & 10 & 17 & 26 & \dots \\
\text{first differe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2451295",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Find all n for which $2^n + 3^n$ is divisible by $7$? I am trying to do this with mods. I know that:
$2^{3k+1} \equiv 1 \pmod 7$, $2^{3k+2} \equiv 4 \pmod 7$, $2^{3k} \equiv 6 \pmod 7$ and $3^{3k+1} \equiv 3 \pmod 7$, $3^{3k+2} \equiv 2 \pmod 7$, $3^{3k} \equiv 6 \pmod 7$, so I thought that the answer would be when $... | $2^{3}\equiv 8\equiv \color{red}1 \bmod 7$, which is why you get a cycle length of $3$ for powers of $2 \bmod 7$. The cycle only closes when you reach $a^k\equiv 1$.
However $3^3=27\equiv 6 \bmod 7$ means that the exponential cycle length for $3\bmod 7$ is more than $3$. Fermat's little theorem says that every cycle l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2454402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
} |
What is $\lim _{ x\rightarrow 0 } \frac { f(x) -x }{ x^2 }$? Given that
$$f(x)=8x-f(3x)-\sin^2(2x),$$
find
$$\lim _{ x\rightarrow 0 } \frac { f\left( x \right) -x }{ x^2 }$$
| $$f(x)=8x-f(3x)-\sin^2(2x)\Longleftrightarrow f(3x) + f(x) = 8x-\sin^2(2x)\\\Longleftrightarrow (f(3x)-3x) + (f(x) -x) = 4x-\sin^2(2x) $$
$$\Longleftrightarrow \frac{f(3x)-3x}{9x^2} + \frac{f(x) -x}{9x^2} = \frac{4x-\sin^2(2x)}{9x^2}$$
Let $$\ell = \lim_{x\to 0}\frac{f(x) -x}{x^2}$$
Then,
$$\lim_{x\to 0}\frac{4x-\sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2455880",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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linear algebra matrix Given the following matrix $A$, find an invertible matrix $U$ so that $UA$ is equal to the reduced row-echelon form of $A$:
$$
A=\begin{bmatrix}
0 & 2 & 2 & 2 \\
0 & 3 & 3 & 3 \\
1 & 1 & -1 & -2
\end{bmatrix}
$$
| *
*Swap rows $1$ and $3$, i.e. pre-multiply by the matrix $\;U_1=\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}$. You get
$$\begin{bmatrix}1&1&-1&-2\\0&2&2&2\\0&3&3&3\end{bmatrix}. $$
*Multiply the (new) second row by $1/2$, i.e. multiply by the matrix $\;U_2=\smash{\begin{bmatrix}1&0&0\\0&\frac12&0\\0&0&1\end{bm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2457986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve $2\log_bx + 2\log_b(1-x) = 4$ I need to solve $$2\log_bx + 2\log_b(1-x) = 4.$$
I have found two ways to solve the problem. The first (and easiest) way is to divide through by $2$:
$$\log_bx + \log_b(1-x) = 2.$$ Then, combine the left side: $$\log_b[x(1-x)] = 2,$$ and convert to the equivalent exponential form, $$... | $\log_b x$ and $\log_b (1-x)$ existing mean $0 < x $ and $0 < 1-x $ so $0 < x < 1$.
When you get $b^2 = x(1-x)$ use similiarity and let $m=|\frac 12 - x|$ so $b^2= (\frac 12 + m)(\frac 12 - m^2) = \frac 14 - m^2 > \frac 14$ so $b < \frac 12$.
Therefore neither solution $x = {1\over2}\left(1\pm\sqrt{1 - 4b^2}\right)$ is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2458109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
How to find the closed form of $f(n) = 9^k \times (-56) + f(n-1)$ I have to find the closed form for
$$T(n) = \begin{cases}
2 , &\text{ if } n=0 \\
9T(n-1) - 56n + 63, &\text{ if } n > 1
\end{cases}$$
I used the repeated substitution method and I found that the pattern for the coefficient of n is equal to the followi... | Usually, these problems are solved using induction. But induction requires you to already 'know' (or guess) the answer. Using generating functions we can solve the problem without having to ever guess:
$$t(n) = 9t(n-1) - 56n + 63$$
$$t(0) = 2$$
Now suppose we write:
$$T(x) = \sum_{n=0}^\infty t(n)x^n$$
We know:
$$T(x) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2463603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
How do you simplify an expression involving fourth and higher order trigonometric functions? The problem is as follows:
Which value of $K$ has to be in order that $R$ becomes independent from $\alpha$?.
$$R=\sin^6\alpha +\cos^6\alpha +K(\sin^4\alpha +\cos^4\alpha )$$
So far I've only come up with the idea that the solu... | Let $x = \sin^2 \alpha,y = \cos^2 \alpha$. Note that $x+y = 1$ so $x^3 + y^3 = (x+y)(x^2+y^2-xy) = (x+y)((x+y)^2-3xy) = 1-3xy$
Also $x^2+y^2 = (x+y)^2-2xy = 1-2xy$
$R = (1-3xy) + K(1-2xy) = 1+K - (3+2K)xy$. So if $K = -3/2$ then $R$ will be independent of $\alpha$. If $K \neq -3/2$ then it will depend on $xy$ which the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2463811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 6,
"answer_id": 0
} |
Solve system of equations: $\sqrt2 \sin x = \sin y, \sqrt2\cos x = \sqrt5\cos y$ $\sqrt2 \sin x = \sin y, \sqrt2\cos x = \sqrt5\cos y$
I tried to use tangent half-angle substitution, tried to sum these two equations and get this
$\sin(x+ \pi/4) = \sqrt3 / \sqrt2 \sin(y+\tan^{-1}(\sqrt5))$
I stuck.
Any Help is appreciat... | Substitute $u=\sin x;\;v=\sin y$. Then square both side of the two equations
$
\left\{
\begin{array}{l}
2 u^2=v^2 \\
2 \left(1-u^2\right)=5 \left(1-v^2\right) \\
\end{array}
\right.
$
Solutions need to be checked. Solution in $[...]$ are not solutions of the given equation
$$\sin x = -\frac{\sqrt{\frac{3}{2}}}{2},\si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2464781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Find area of triangle ABD Let ABC be a right-angled triangle $B=90$. Let in a triangle pick a point $D$ inside the triangele such that $AD= 20, DC=15, DB=10$ and $AB=2BC$. What is the area of triangle $ABD$?
Thanks!
| i have got three equations
$$5a^2=20^2+15^2-2\cdot 20\cdot 15\cos(2\pi-\alpha-\beta)$$
$$a^2=15^2+10^2-2\cdot 15\cdot 10\cos(\beta)$$
$$4a^2=10^2+20^2-2\cdot 10\cdot20\cos(\alpha)$$
solving this we get $$\alpha=\arctan\left(\frac{1}{2}\right)$$
and $$A_{\Delta ABD}=\frac{1}{2}20\cdot 10\sin\left(\arctan\left(\frac{1}{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2467383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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How to find $\lim_{x \to 1^-} (x+1) \lfloor \frac{1}{x+1}\rfloor $? find the limits :
$$\lim_{x \to 1^-} (x+1) \lfloor \frac{1}{x+1}\rfloor =?$$
My try :
$$\lfloor \frac{1}{x+1}\rfloor=\frac{1}{x+1}-p_x \ \ \ : 0\leq p_x <1$$
So we have :
$$\lim_{x \to 1^-} (x+1) (\frac{1}{x+1}-p_x) \\= \lim_{x \to 1^-} -(x+1)p_x=!??$... | $$\lim_{x \to 1^-} (x+1) \left\lfloor \frac{1}{x+1}\right\rfloor $$
Let ${x = 1-y}$
So our equation is same as
$$\lim_{y \to 0^+} (2-y) \left\lfloor \frac{1}{2-y}\right\rfloor $$
Floor value is $0$. Also value outside floor funtion tends to $2$.
Thus our equation becomes
$2\cdot 0=0$
So answer is $0$.
QED
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2475074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Proving that: $ | a + b | + |a-b| \ge|a| + |b|$ I am trying to prove this for nearly an hour now:
$$
\tag{$\forall a,b \in \mathbb{R}$}| a + b | + |a-b| \ge|a| + |b|
$$
I'm lost, could you guys give me a tip from where to start, or maybe show a good resource for beginners in proofs ?
Thanks in advance.
| Here's the way I see this geometrically in $\mathbb{C}$: let's say we have two complex numbers $a, b$ and consider the parallelogram formed by $0, a, a + b, b$. The midpoints of the diagonals coincide at the point $\frac{a + b}{2}$. These diagonals cut the parallelogram into four triangles, on each of which we can perf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2477064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 9,
"answer_id": 5
} |
Integrate $\arctan{\sqrt{\frac{1+x}{1-x}}}$ I use partial integration by letting $f(x)=1$ and $g(x)=\arctan{\sqrt{\frac{1+x}{1-x}}}.$ Using the formula:
$$\int f(x)g(x)dx=F(x)g(x)-\int F(x)g'(x)dx,$$
I get
$$\int1\cdot\arctan{\sqrt{\frac{1+x}{1-x}}}dx=x\arctan{\sqrt{\frac{1+x}{1-x}}}-\int\underbrace{x\left(\arctan{\sq... | Inverse trig identities are less familiar than regular trig identities. So perhaps if you invert this, so that regular trig identities would apply, you find something noteworthy:
$$\begin{align}
y&=\arctan\sqrt{\frac{1+x}{1-x}}\\
\tan(y)&=\sqrt{\frac{1+x}{1-x}}\\
\tan^2(y)&=\frac{1+x}{1-x}\\
\tan^2(y)-x\tan^2(y)&=1+x\\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2477162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
Find $\alpha + \beta$ given that: $\alpha^3-6\alpha^2+13\alpha=1$ and $\beta^3-6\beta^2+13\beta=19$ , ( $\alpha , \beta \in \Bbb R)$
Find $\alpha + \beta$ given that: $\alpha^3-6\alpha^2+13\alpha=1$ and $\beta^3-6\beta^2+13\beta=19$ , ( $\alpha , \beta \in \Bbb R)$
I have a solution involving a variable change ($\alp... | Adding both the equations we have $$\alpha^3-6\alpha^2+13\alpha+\beta^3-6\beta^2+13\beta=20$$
Which is equivalent to
$$(\alpha-2)^3+(\beta-2)^3+(\alpha-2)+(\beta-2)=0$$
Let $\alpha-2=a$ and $\beta -2=b$
Now we've ;
$$a^3+b^3+a+b=0$$
$$(a+b)(a^2-ab+b^2+1)=0$$
This give us
$$a+b=0 \implies \color{blue}{\alpha+\beta=4}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2481243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Find all real solutions of $\sqrt{x} - \sqrt{2-2x} = 1$ Squaring both sides of $\sqrt{x} - \sqrt{2-2x} = 1$ and rearranging
I arrive at the quadratic $9x^2 - 10x + 1 = 0$ which has solutions $x=1/9$ and $x=1$.
I don't understand why $x=1$ fits the original equation but $x=1/9$ doesn't (left hand side gives $-1$).
| $$\sqrt{x} - 1 = \sqrt{2-2x}$$
$$x -2\sqrt{x} + 1 = 2-2x$$
$$3x -2\sqrt{x} - 1 = 0$$
Now if $\sqrt{x} = u$ we have:
$$3u^2 -2u - 1 = 0$$
$$u = 1 \vee u=-\frac{1}{3}$$
In the case $u = 1$ we've found the solution $x = 1$. But in the case $u = -\dfrac{1}{3}$ we get:
$$\sqrt{x} = -\frac{1}{3}$$
Do you see the issue now?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2481743",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Center of Mass Help For a question, it asked us to find the center of mass of the region between $y=\sqrt x$ and $y=x^3$. I was wondering if the way that I did it was correct. If not, what is the correct formuala for solving center of mass Center of Mass
| The center of mass would be located by using the following formulas:
$\bar{x}=\frac{1}{A}\int_\limits{a}^bx[f(x)-g(x)]dx$ and $\bar{y}=\frac{1}{A}\int_\limits{a}^b\frac{1}{2}[(f(x))^2-(g(x))^2]dx$
$A=\int_\limits{a}^b[f(x)-g(x)]dx$
$\bar{x}=\frac{\int_\limits{0}^1x(\sqrt{x}-x^3)dx}{\int_\limits{0}^1\sqrt{x}-x^3dx}=\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2482225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Finding the value of a tricky limit I have some difficulties with my homework in mathematical analysis and I don't really have any good ideas, how to get off the mark. If you, guys, could give me any ideas, tips or solutions for the following task, I would be really thankful.
The task is as follows:
Let $a_{n}$ be ... | Trying the problem with $a_n=c$ suggests the answer is $0$.
Since $\limsup\limits_{n}\frac{a_1 + a_2 + ... + a_n }{n} <\infty$, there exists some $M>0$ such that for all $n$, $\displaystyle \frac{a_1 + a_2 + ... + a_n }{n}\leq M$.
Let $\epsilon>0$. Since $\lim_n \frac{a_n}n=0$, there is some $N$ such that $n\geq N\imp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2482718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Finding value of determinant, Where $A+B+C+P+Q+R=0$ Finding value of Determinant $$\begin{vmatrix}
\tan (A+P) & \tan(B+P) & \tan(C+P)\\\\
\tan (A+Q)& \tan (B+Q) & \tan (C+Q)\\\\
\tan (A+R)&\tan (B+R) & \tan(C+R)
\end{vmatrix}$$ for all values of $A,B,C,P,Q,R$ Where $A+B+C+P+Q+R=0$
I did not understand how to st... | We want to find the determinant of this Matrix.
$$
\begin{pmatrix}
Tan(A+P) & Tan(B+P) & Tan(C+P)\\
Tan(A+Q) & Tan(B+Q) & Tan(C+Q)\\
Tan(A+R) & Tan(B+R) & Tan(C+R)
\end{pmatrix}
$$
The determinant of this matrix gives us \to
$$
-Tan(A+R) \cdot Tan(B+Q) \cdot Tan(C+P)\\
+Tan(A+Q) \cdot Tan(B+R) \cdot Tan(C+P)\\
+Tan(A+R... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2484263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Determine the sum of function series The serie is :
$$\sum _{n=1}^{\infty }{\frac { \left( -1 \right) ^{n} \left( 2\,n-1
\right) {x}^{2\,n}}{{4}^{n}}}
$$
I found out the derivative $2\,{\frac {n \left( -1 \right) ^{n}{\left( 2\,n-1 \right)}{x}^{2\,n-1}
}{{4}^{n}}}
$ , which gave me this $2\,n-1$ in power and in num... | $$\sum _{n=1}^{\infty }{\frac { \left( -1 \right) ^{n} \left( 2\,n-1
\right) {x}^{2\,n}}{{4}^{n}}}=\sum _{n=1}^{\infty } \frac{(-1)^n (2 n) x^{2 n}}{4^n} + \sum _{n=1}^{\infty } \frac{(-1)^{n+1} x^{2 n}}{4^n}$$
For the first series
$$\sum _{n=1}^{\infty } \frac{(-1)^n x^{2 n}}{4^n}=\sum _{n=1}^{\infty } \left(-\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2487998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
Numbers $a^{p}a^q$ congruent to 1 modulo pq Consider numbers of the form $a^{p}a^q=a^{p+q} $ where $p,q$ are primes.
Now I'm interested when such number is congruent to $1$ modulo product $pq$.
Take for example $p=3$ and $q=5$.
Then it seems that every number $a$ coprime with $p$ and $q$ is congruent to $1$ modulo $(... | This is a partial answer.
*
*Why it holds for $(3,5)$ and $(5,7)$, ... for others probably not?
Let us prove that $a^{3+5}\equiv 1\pmod{3\times 5}$ if $\gcd(a,3\times 5)=1$.
We have$$a^{3+5}-1=a^8-1=(a-1)(a+1)(a^2+1)(a^4+1)$$
*
*If $a\equiv 1\pmod 3$, then $a-1\equiv 0\pmod 3$
*If $a\equiv 2\pmod 3$, the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2496941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Solve for x: $2^x+4^x=8^x$ I tried turning $4^x$ and $8^x$ into powers of $2$ and manipulating the equation but could not make progress. What would be the next steps for this problem?
| Since $4^{x} = 2^{2x}$ and $8^{x} = 2^{3x}$ then the equation becomes
$$2^{x} + 2^{2 x} = 2^{3 x}.$$
Let $t = 2^{x}$ to obtain $t^{3} - t^{2} - t = 0$ or $t (t^{2} - t - 1) = 0$. The quadratic can be factored using $2 \alpha = 1 + \sqrt{5}$ and $2 \beta = 1 - \sqrt{5}$ for which
$$t \, (t - \alpha) \, (t - \beta) = 0.$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2499158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Find Probability that even numbered face occurs odd number of times A Die is tossed $2n+1$ times. Find Probability that even numbered face occurs odd number of times
First i assumed let $a,b,c,d,e,f$ be number of times $1$ occurred, $2$ occurred and so on $6$ occurred respectively. Then we have
$$a+b+c+d+e+f=2n+1$$ i... | Even and odd faces both occur with a proabability of $\frac{1}{2}$. The probability of attaining an odd number of evens is
\begin{eqnarray*}
\sum_{i=0}^{n} \binom{2n+1}{2i+1} \left( \frac{1}{2} \right)^{2n+1}
\end{eqnarray*}
and an even number of odd faces is
\begin{eqnarray*}
\sum_{i=0}^{n} \binom{2n+1}{2i} \left( \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2502305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
let $f(\frac{x}{3})+f(\frac{2}{x})=\frac{4}{x^2}+\frac{x^2}{9}-2$ then find $f(x)$ let $f(\frac{x}{3})+f(\frac{2}{x})=\frac{4}{x^2}+\frac{x^2}{9}-2$
then find $f(x)$
My Try :
$$f(\frac{x}{3})+f(\frac{2}{x})=(\frac{2}{x})^2-1+(\frac{x}{3})^2-1$$
So we have :
$$f(x)=x^2-1$$
it is right ?Is there another answer?
| A particular solution is $f(x)=x^2-1$. The general solution of the associated homogeneous problem
$$f\left({x\over3}\right)+f\left({2\over x}\right)=0$$
is $$f_{\rm hom}(x)=u\left(\log\bigl(\sqrt{3/2}\> x\bigr)\right)\qquad(x>0)\ ,$$
whereby $u$ is an arbitrary odd function. It follows that the general solution of the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2502655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Prove that: $1-\frac 12+\frac 13-\frac 14+...+\frac {1}{199}- \frac{1}{200}=\frac{1}{101}+\frac{1}{102}+...+\frac{1}{200}$ Prove that:
$$1-\frac 12+\frac 13-\frac 14+...+\frac {1}{199}- \frac{1}{200}=\frac{1}{101}+\frac{1}{102}+...+\frac{1}{200}$$
I know only this method:
$\frac {1}{1×2}+\frac {1}{2×3}+\frac {1}{3×4... | We have
\begin{split} &&1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots + \frac{1}{199} - \frac{1}{200} \\&=&
\left(1 + \frac{1}{3} + \frac{1}{5} +\cdots + \frac{1}{197}+\frac{1}{199} \right) - \left(\frac{1}{2} + \frac{1}{4} + \cdots + \frac{1}{200}\right)\\
&=&
\left(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2502982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 3
} |
Explicit sum of $\sum_{n=0}^{\infty} \frac{6+3^n}{6^{n+2}}$ I want to find the sum of the infinite series $$\sum_{n=0}^{\infty} \frac{6+3^n}{6^{n+2}}$$
So far I have managed to simplify the expression to $$\sum_{n=0}^{\infty} \left(\frac {1}{6^{n+1}} + \frac{1}{9\cdot2^{n+2}}\right)$$.
The sum clearly converges as both... | As mentioned,it is geometric series $$\sum _{ n=0 }^{ \infty } \left( \frac { 1 }{ 6^{ n+1 } } +\frac { 1 }{ 9\cdot 2^{ n+2 } } \right) =\frac { 1 }{ 6 } \sum _{ n=0 }^{ \infty } \frac { 1 }{ 6^{ n } } +\frac { 1 }{ 36 } \sum _{ n=0 }^{ \infty } \frac { 1 }{ 2^{ n } } =\\ =\frac { 1 }{ 6 } \left( 1+\frac { 1 }{ 6 }... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2503207",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
coefficient of $x^{17}$ in $(x^2+x^3+x^4+x^5+x^6+x^7)^3$ coefficient of $x^{17}$ in $(x^2+x^3+x^4+x^5+x^6+x^7)^3$ - I'm reviewing for an exam and don't understand the answer, C(11+3−1,11)−C(3,1)×C(5+3−1,5).
| $$
\begin{align}
\left[x^{17}\right]\left(x^2+x^3+x^4+x^5+x^6+x^7\right)^3
&=\left[x^{11}\right]\left(1+x+x^2+x^3+x^4+x^5\right)^3\\
&=\left[x^{11}\right]\left(\frac{1-x^6}{1-x}\right)^3\\
&=\left[x^{11}\right]\sum_{j=0}^3(-1)^j\binom{3}{j}x^{6j}
\sum_{k=0}^\infty(-1)^k\binom{-3}{k}x^k\\
&=\left[x^{11}\right]\sum_{j=0}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2504632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Limit of a sequence by definition I need to prove by definition that the limit of sequence :
$$a_n = \frac{5n^3-3n^2+1}{4n^3+n+2}$$
is $\dfrac54$ which means I need to show that :
$$\left|\frac{5n^3-3n^2+1}{4n^3+n+2} - \frac54\right| < \varepsilon$$
I tried for hours to solve it but could not,
Can anyone help please?
T... | Let $\varepsilon > 0$. Pick $n_0 \in \mathbb{N}$ such that $n_0 > \frac{23}{16\varepsilon}$.
For $n \ge n_0$ we have:
\begin{align}
\left|\frac{5n^3-3n^2+1}{4n^3+n+2} - \frac54\right| &= \left|\frac{4(5n^3-3n^2+1)-5(4n^3+n+2)}{4(4n^3+n+2)}\right|\\
&= \left|\frac{-12n^2-5n-6}{4(4n^3+n+2)}\right|\\
&= \frac{12n^2+5n+6}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2506052",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove identity composed of floor functions How to prove:
$$\left\lfloor \frac{x+2}{6}\right\rfloor -\left\lfloor \frac{x+3}{6}\right\rfloor +\left\lfloor \frac{x+4}{6}\right\rfloor =\left\lfloor \frac{x}{2}\right\rfloor -\left\lfloor \frac{x}{3}\right\rfloor$$
where $x\in \mathbb{R}$.
I think it is problem made by Rama... | Hint: write $x= 6k+r$ where $o\leq r <6$ and $k\in \mathbb{Z}$. Then we have:
$$\left\lfloor \frac{x+2}{6}\right\rfloor -\left\lfloor \frac{x+3}{6}\right\rfloor +\left\lfloor \frac{x+4}{6}\right\rfloor =
k + \underbrace{\left\lfloor \frac{r+2}{6}\right\rfloor -\left\lfloor
\frac{r+3}{6}\right\rfloor +\left\lfloor
\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2506218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Proving $4^n=\sum_{k=0}^n2^k\binom{2n-k}{n}$ $4^n = \sum\limits_{k=0}^{n}2^k\cdot{{2n - k} \choose n}$
I tried formal power series, but failed.
| With formal power series as requested by OP we may write
$$\sum_{k=0}^n 2^k {2n-k\choose n}
= 2^n \sum_{k=0}^n 2^{-k} {n+k\choose n}
= 2^n \sum_{k=0}^n 2^{-k} [z^n] (1+z)^{n+k}
\\ = 2^n [z^n] (1+z)^n \sum_{k=0}^n 2^{-k} (1+z)^k
= 2^n [z^n] (1+z)^n \frac{1-(1+z)^{n+1}/2^{n+1}}{1-(1+z)/2}
\\ = 2^{n+1} [z^n] (1+z)^n \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2507702",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 2
} |
Is there a good upper bound for $(x-1)^n-x^n$ for $x\ge 1$ and $n=2,3,4,...$? For an positive integer $n\ge 2$, is there a good upper bound for $(x-1)^n-x^n$ for $x> 1$?
Revised question: Is it less than $-\log(x-1)$ for all $x>1$?
By Binomial theorem, $(x-1)^n-x^n=\sum_{k=1}^n\binom{n}{k}x^{n-k}(-1)^k$. Or by Mean Va... | The upper bound $-n\zeta^{n-1} \le -n(x-1)^{n-1}$ is certainly less that $-\log(x-1)$ for $x \ge 2.
You can get better upper bounds by using Taylor approximations of $f(t) = t^n$ about $t = x$ and Lagrange's remainder term. For example:
$$
(x-1)^n-x^n = -nx^{n-1} + \frac{1}{2} n(n-1)\zeta^{n-2} \le
-nx^{n-1} + \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2511699",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Muller's recurrence: limit In Kahan's account of the Muller's recurrence (p.16): $x_{n+1} = E(x_n, x_{n-1})$ for the function
$$
E(y, z) = 108 - \frac{815-\frac{1500}{z}}{y},
$$
he uses the characteristic polynomial for the recurrence to deduce the closed form:
$$
x_n = \frac{\alpha 3^{n+1} + \beta 5^{n+1} + \gamma 100... | $\alpha = 1, \beta = 1, \gamma = 0$ gives the exact solution from the starting values. For the round-off error setting they are slightly disturbed; we still have
$\alpha \approx 1, \beta \approx 1,$ but $\gamma$ has the order of the machine epsilon $\gamma \approx \epsilon.$ To compute Kahan's form from
$$x_n = \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2513965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Prove that $\cosh(x)=\sec(\theta )$ if $x=\ln(\sec \theta + \tan \theta)$ I'm trying to prove that $\cosh(x)=\sec(\theta )$ if $x=x=\ln(\sec \theta + \tan \theta)$. I've substituted the value of $x$ into $\cosh(x)$ to get $$\frac{e^{\ln(\sec \theta + \tan \theta)}+e^{-\ln(\sec \theta + \tan \theta)}}{2}$$ and simplifie... | You got to
$$
\cosh x = \frac{1}{2}\left(\sec\theta + \tan\theta + \frac{1}{\sec\theta+\tan\theta}\right)
$$
Write this is as single fraction:
$$
\frac{1}{2}\left(\sec\theta + \tan\theta + \frac{1}{\sec\theta+\tan\theta}\right)
= \frac{(\sec\theta + \tan\theta)^2 + 1}{2(\sec\theta + \tan\theta)}
$$
Expand and remem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2514291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
an optimisation problem folding paper
A standard $8.5$ inches by $11$ inches piece of paper is folded so that one corner touches the opposite long side and the crease runs from the adjacent short side to the other long side, as shown in the picture below. What is the minimum length of the crease?
So I put the shor... |
Perhaps, a simpler approach
would be
to express $|EG|$ in terms of
$\angle ABF=\phi$.
In this case
\begin{align}
|BF|&=\frac{w}{\cos\phi}
,\\
|BK|&=\tfrac12\,\frac{w}{\cos\phi}
,\\
|BE|&=\frac{|BK|}{\cos(\tfrac\pi2-\phi)}
=\frac{w}{2\,\cos\phi\sin\phi}
,\\
|EG|&=\frac{|BE|}{\cos\phi}
=\tfrac12\,\frac{w}{\cos^2\ph... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2516662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Proving that limit (without using calculus) $(n^2 + n) ^{1/2} - n = \frac{1}{2}$ Prove, Without using Calculus.
$$\lim_{n->\infty} \left((n^2 + n)^{1/2} - n\right) = \frac{1}{2} $$
I am taking an introduction to real analysis course going through Rudin.
My work so far,
By definition, limit of a sequence $p$ has the ... | Yep it's essentially the right approach. We want to find $N$ so that if $n > N$, then
$$ \left|\left((n^2 + n)^{1/2} - n\right) - \frac{1}{2} \right| < \varepsilon $$
We know that
$\left((n^2+n)^{1/2} - (n + 1/2)\right)\left((n^2+n)^{1/2} + (n+1/2)\right) = (n^2+n)-(n+1/2)^2$ using a "difference of two squares". Hence... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2517379",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Residue of $f(z) = \frac{1}{z(1-\cos(z))}$ I tried to find the residue of the function $$f(z) = \frac{1}{z(1-\cos(z))}$$ at the $z=0$. So I did:
\begin{align}
\hbox{Res}(0)&=\lim_{x \to 0}(z)(f(z))\\
\hbox{Res}(0)&= \lim_{x \to 0} \frac{1}{1-\cos(z)}
\end{align}
and I got that the residue to be $\infty$, and it seems t... | $$\cos z=1-\dfrac{1}{2!}z^2+\dfrac{1}{4!}z^4-\dfrac{1}{6!}z^6+\cdots$$
then
\begin{align}
\dfrac{1}{z(1-\cos z)}
&= \dfrac{1}{z^3\left(\dfrac{1}{2!}-\dfrac{1}{4!}z^2+\dfrac{1}{6!}z^4+\cdots\right)} \\
&= \dfrac{1}{z^3}\left(2+\dfrac{1}{6}z^2+\dfrac{1}{120}z^4+\cdots\right) \\
&= \dfrac{2}{z^3}+\dfrac{1}{6z}+\dfrac{z}{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2517601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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A problem :$(edf)^3(a+b+c+d+e+f)^3\geq(abcdef)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{1}{f})^3$ Hello during a problem I have this to solve :
Let $a,b,c,d,e,f$ be real positiv number such that $a\geq b\geq c\geq d\geq e\geq f\geq 1$ then :
$$(edf)^3(a+b+c+d+e+f)^3\geq(abcdef)(\frac{1}{a}+\fra... | This inequality simply follows from the original inequality for $n=3$, so I guess its proof won't be easier. Indeed, dividing both sides by $def$ we transform it to
$$(def)^2(a+b+c+d+e+f)^3\geq abc\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{1}{f}\right)^3.$$
Now we note that with $a$, $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2519854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Simplifying expressions. How do you simplify the following expression?
$$Q=(1-\tan^2(x)) \left(1-\tan^2 \left(\frac{x}{2}\right)\right)\cdots \left(1-\tan^2\left(\frac{x}{2^n}\right)\right)$$
I've tried the the following :
$\tan x = \frac{\sin x}{\cos x}$
$$\cos x = \frac{\sin 2x}{2\sin x}$$
$$\tan x = \frac{2\sin^2x}{... | Hint:
$$
1-\tan^2 t=\frac{\cos^2t-\sin^2t}{\cos^2t}=\frac{\cos(2t)}{\cos^2t}
$$
and for arguments of the form $\frac{x}{2^n}$, many things will vanish in the product.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2520516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Maximum possible value of $P(10)$ The real numbers $a$, $b$, $c$, and $d$ are each less than or equal to $12$. The polynomial $$P(x)=ax^3+bx^2+cx+d$$ satisfies $P(2)=2$, $P(4)=4$ and $P(6)=6$. Find the maximum possible value of $P(10)$.
What I did was first I used the given information to get $3$ equation in $a, b,c,d... | $P(x)=x+a(x-2)(x-4)(x-6)$ because $2,4,6$ are roots of the cubic $P(x)-x$, whose leading coefficient is $a$.
Therefore, $P(x)=a x^3 - 12 a x^2 + (44 a + 1) x - 48 a$.
Since the coefficients are all at most $12$, we need
$$
a \le 12, \quad -12 a \le 12, \quad 44a+1 \le 12, \quad -48a \le 12
$$
Therefore,
$
-\frac14 \le ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2521862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Find $\lim_{n\rightarrow\infty} \frac{(n+1)^{2n^2+2n+1}}{(n+2)^{n^2+2n+1} n^{n^2}}$ Woflram gives $\frac{1}{e}$ as the limit, but I failed to obtain it. Please help.
$$\lim_{n\rightarrow\infty} \frac{(n+1)^{2n^2+2n+1}}{(n+2)^{n^2+2n+1}n^{n^2}}$$
| \begin{align}
\lim_{n\rightarrow\infty} \frac{(n+1)^{2n^2+2n+1}}{(n+2)^{n^2+2n+1}n^{n^2}}
&=\lim_{n\rightarrow\infty} \frac{(n+1)^{2n^2}(n+1)^{2n+1}}{n^{n^2}(n+2)^{n^2}(n+2)^{2n+1}}\\
&=\lim_{n\rightarrow\infty} \frac{(n+1)^{2n^2}(n+1)^{2n}(n+1)}{\left[n(n+2)\right]^{n^2}(n+2)^{2n}(n+2)}\\
&=\lim_{n\rightarrow\infty} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2522605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Number of solutions to $a + b + c + d + f = n$ in nonnegative integers $a, b, c, d, f,$ given restrictions on $a, b,$ and $f$ Count the number of solutions to $a + b + c + d + f = n$ in nonnegative integers $a, b, c, d, f,$ such that $a$ is a multiple of 4, $b$ is at most 1, and $f$ is either 0 or 2.
My attempt:
As far... | The generating function is
\begin{eqnarray*}
\underbrace{\frac{1}{(1-x^4)}}_{a} \underbrace{(1+x)}_{b} \underbrace{\frac{1}{(1-x)^2}}_{c \text{ and } d } \underbrace{(1+x^2)}_{f}.
\end{eqnarray*}
Which can be simplified to
\begin{eqnarray*}
\color{blue}{\frac{1}{(1-x)^3}}.
\end{eqnarray*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2523676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Find the$A:=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...\frac{1}{20^2}$
Find the :
$$A:=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...\frac{1}{20^2}$$
My Try :
$$\frac{1}{2^2}+\frac{1}{3^2}=\frac{3^2+2^2}{2^2\cdot 3^2}$$
$$\frac{3^2+2^2}{2^2\cdot 3^2}+\frac{1}{4^2}=\frac{(3^2+2^2)(4^2+1)}{4^2\cdot 3^2\cdot 2^2}$$
$$... | We know that the sum of the reciprocal of the squares add up to $\pi^2/6.$ So your sum equals
$$\frac{\pi^2}{6} -1 -\sum_{n=21}^{\infty} \frac{1}{n^2}.$$
The infinite sum is less than, but very close to
$$\int_{20}^\infty \frac{1}{x^2} \; dx = \frac{1}{20}.$$
So your sum is about
$$\frac{\pi^2}{6} -1 -\frac{1}{20} = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2526394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How to evaulate $\int_0^{\infty}\frac{\ln x}{\sqrt{x}(x^2+a^2)^2}$ using contour integration I'm asked to calculate the following integral for which $0 \neq a \in \mathbb{R}$:
$$\int_0^{\infty}\frac{\ln x}{\sqrt{x}(x^2+a^2)^2}$$
I'm confused about which contour I should use, whether it should be a semi-circle deformed ... | We integrate with $a$ a positive real
$$f(z) = \frac{\mathrm{Log}(z)}{(z-ai)^2 (z+ai)^2}
\exp(-1/2\times \mathrm{Log}(z))$$
around a keyhole contour with the slot on the positive real axis,
which is also where the branch cut of the logarithm is located
(argument of the logarithm is between $0$ and $2\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2527231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Solve $X^2=A$, where X is a 2 by 2 matrix and A is a known matrix A = $
\begin{pmatrix}
1 & 1 \\
-1 & 1 \\
\end{pmatrix}
$
. I wrote X = $
\begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix}
$. So $X^2$=$
\begin{pmatrix}
a^2+bc & ab+bd \\
... | You get:
$a^2 +bc$ = 1
$ac + bc$ = -1
$ab + bd$ = 1
$bc + d^2$ = 1
Hence use substitution to find a, b, c and d.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2527541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Taylor inside an integral I know the following integral should be:
$$ \int_{0}^{1} \frac{dx}{\sqrt{1-x^2-\epsilon(1-x^3)}} \approx \pi/2 + \epsilon$$
for $\epsilon$ small. What I do is set $-\epsilon(1-x^3)=y$ and then since $y$ it's always greatly smaller than $(1-x^2)$ I do Taylor expansion around $y=0$:
$$ \frac{dx... | You can be authorized in doing that because $\epsilon$ is small (we suppose small enough to give you the "license" of dong that).
Also notice that the range of the integral is $[0, 1]$, hence $x$ too, in a certain way, is small, and terms like $x^3$ are then even smaller.
There are many approaches for dealing with suc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2529544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
How to show the action of $SL_2(\mathbb R)$ on the complex upper half plane is transitive? I've seen other answers here and here but I still am not understanding how to show that this action is transitive. I want to show that for all $z \in \mathbb H$ there exists a matrix $g \in SL_2(\mathbb R)$ such that $gi=z$.
If I... | ANSWER:
If $A \in SL(2,\mathbb R)$, then
\begin{align*}
A\cdot i&=\frac{ai+b}{ci+d}\\
&=\frac{ai+b}{ci+d}\cdot \frac{(-ci+d)}{(-ci+d)}\\
&=\frac{ac+bd+i(ad-bc)}{c^2+d^2}.
\end{align*}
We're in $SL(2, \mathbb R$), so $ad-bc=1$, and
$$\frac{ac+bd+i(ad-bc)}{c^2+d^2}=\frac{ac+bd}{c^2+d^2}+i\frac{1}{c^2+d^2}.$$
If $c=0$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2529664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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How to prove this Fibonacci identity? $\sum_{k=0}^{n} F_{k} F_{n-k} = \frac{1}{5}\left(n L_{n} - F_{n}\right)$ How to prove this Fibonacci identity?
$$\sum_{k=0}^{n-3} F_{k} F_{n-k-3} = \frac{(n-3)L_{n-3} - F_{n-3}}{5}$$
i tried to used the generating function and partial decomposition but i got confused?
| Using
\begin{align}
F_{n} &= \frac{\alpha^{n} - \beta^{n}}{\alpha - \beta} \\
L_{n} &= \alpha^{n} + \beta^{n} \\
2 \alpha &= 1 + \sqrt{5} \\
2 \beta &= 1 - \sqrt{5}
\end{align}
then
\begin{align}
(\alpha - \beta)^{2} \, F_{n} F_{n-k} &= L_{n} - \beta^{n} \, \left(\frac{\alpha}{\beta}\right)^{k} - \alpha^{n} \, \left(\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2531845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Integration contour for improper integral $\int^\infty_{-\infty} \frac{dx}{(e^x + 1)(1 + e^{-x-z})}$. A while ago, I read through a book in solid state physics (Ziman's Electrons and Photons) and, as an algebraic step in the derivation of a formula, the author uses
$$
\int_{-\infty}^{\infty}\frac{d\eta}{\{\exp\eta + 1... | The integral is indeed elementary.
For initial notational ease, let $a = e^{-z}$ and we will consider the indefinite integral
$$I = \int \frac{dx}{(e^x + 1)(1 + a e^{-x})}.$$
Observing that
$$\frac{1}{(e^x + 1)(1 + a e^{-x})} = \frac{1}{1 -a} \left [\frac{1}{e^x + 1} - \frac{a e^{-x}}{1 + a e^{-x}} \right ],$$
the inte... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2531962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Determine $\lim_{n \to \infty} (2^{n+1} \sin(\frac{\pi}{2^{n+1}}))$ I was able to calculate
$$\lim_{n \to \infty} (2^{n+1} \sin(\frac{\pi}{2^{n+1}})) = \pi$$
using L'Hopital Rule, but how to get $\pi$ without it?
| since $\sin \left( \frac { \pi }{ 2^{ n+1 } } \right) \overset { n\rightarrow \infty }{ \longrightarrow } 0$ we can apply famous limit $\lim _{ n\rightarrow 0 }{ \frac { \sin { n } }{ n } } =1$
$$\lim _{ n\to \infty } \left( 2^{ n+1 }\sin \left( \frac { \pi }{ 2^{ n+1 } } \right) \right) =\lim _{ n\to \inf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2534330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Triple integration of a solid bounded by 2 curves $z=1-x^2-2y^2$ and $z=x^2$.
I need to find the Vol bounded by $z=1-x^2-2y^2$ and $z=x^2$.
I have gotten
$$V=∫∫∫rdrdθdz$$
where the limits are $0$ to $2\pi$, $0$ to $\sqrt{1/2}$, and $0$ to $1$.
The ans is $\pi$ but my ans is $\pi/2$. Can anyone tell me where my mista... | $ z = 1 - x^2 - 2y^2$ and $z = x^2 $ meet along the circle $x^2+y^2=\frac{1}{2} $
So, limits of $ x$ would be $ \frac{-1}{\sqrt{2}}<x<\frac{1}{\sqrt{2}}$ and
$y$ would be $ -\sqrt{\frac{1}{2}-x^2}<y<\sqrt{\frac{1}{2}-x^2}$ and
$z$ would be $x^2<z<1-x^2-2y^2$
So,
$V=\int_{x^2}^{1-x^2-2y^2} \int_{-\sqrt{\frac{1}{2}-x^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2536216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Trigonometric sum of a ratio of two sine functions
Evaluate $$\sum^{13}_{k=1}\frac{\sin (30^\circ k +45^\circ)}{\sin(30^\circ(k-1)+45^\circ)}.$$
Put $30^\circ = \alpha, 45^\circ = \beta$. Then
$$\begin{align}
S:=&\sum^{13}_{k=1}\frac{\sin (\alpha k+\beta)}{\sin(\alpha(k-1)+\beta)} = \sum^{13}_{k=1}\frac{\sin(\alpha(k... | You are on the right track. Note that $\cot(x)=\tan(90^{\circ}-x)$ and $\tan(x)=\tan(x+180^{\circ})$. Hence
$$\begin{align}\sum_{k=1}^{13}&\cot((k-1)30^\circ+45^\circ)=\sum_{k=0}^{12}\tan(45^\circ-k30^\circ)\\
&=1+\sum_{k=1}^{6}\tan(45^\circ-k30^\circ)+\sum_{k=1}^{6}\tan(45^\circ-(k+6)30^\circ)\\
&=1+2\sum_{k=1}^{6}\ta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2537835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Find min of $\int_{-1}^1| x^3-a-bx-cx^2|^2 dx$ and max of $\int_{-1}^1 x^3 g(x) dx$
1) Apply best approximation theorem to compute min$\int_{-1}^1| x^3-a-bx-cx^2|^2 dx$.
2) Find max $\int_{-1}^1 x^3 g(x) dx$, where $g$ is subject to the constraints
$$\int_{-1}^1 g(x)dx = \int_{-1}^1xg(x)dx=\int_{-1}^1 x^2 g(x) dx=0... | Hint for 1). Since $\int_{-1}^1x^d dx=0$ when $d$ is odd, and $\int_{-1}^1x^d dx=\frac{2}{d+1}$ when $d$ is even, by expanding the square we get
$$\int_{-1}^1(x^3-a-bx-cx^2)^2 dx
=\frac{2}{3}b^2-\frac{4}{5}b+\frac{2}{7}+2a^2+\frac{4}{3}ac+\frac{2}{5}c^2\\
=\frac{2}{3}\left(b-\frac{3}{5}\right)^2+\frac{8}{175}+2\left(a^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2539486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Solve the integral $\int_0^1 x^{\frac{1}{3}}(1-x)^{\frac{2}{3}}dx$ I would like to consider two ways to compute the (real) integral $\int_{0}^{1}x^{\frac{1}{3}}(1-x)^{\frac{2}{3}}dx$ using complex analysis:
(i) By residues
(ii) By Beta function
My computations:
(i) First of all, my intuition is that the value of the i... | You seem to want to use Cauchy's residue theorem (not to be confused with the Cauchy-Riemann equations, BTW) to evaluate this integral. To use the residue theorem, you need to integrate over a closed contour in the complex plane. As far as I can tell, you don't have one; you're just adding up all the residues that y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2539559",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find the result of the root expression, is my answer correct or not? Suppose $a < 0 < b$. Then what is the result of:
$\sqrt{(a-b)^2} + \sqrt[6]{ b^6 } = ?$
I have a solution but I can't be sure if I did a mistake, because I usually do! My solution:
Call $a = -c$ for some $0 < c $. Then,
$=\sqrt{(-c-b)^2} + \sqrt[6]{ ... | i would write $$\sqrt{(a-b)^2}+\sqrt[6]{b^6}=\sqrt{(b-a)^2}+\sqrt[6]{b^6}=b-a+b$$
this is the same as you
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2539919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Can this definite integral involving series be solved without a calculator? I got this question today but I can't see if there is any good way to solve it by hand.
Evaluate the definite integral
$$\int_2^{12}\frac{\sqrt{x+\sqrt{x+\sqrt{x+...}}}}{\sqrt{x\sqrt{x\sqrt{x}...}}}\,\mathrm{d}x$$
where the series in the numer... | For 2. notice that
$$\int_{2}^{12}\frac{\sqrt{1+4x}}{2x}dx=\int_{2}^{12}\frac{1+4x}{2x\sqrt{1+4x}}=\frac{1}{2}\left(\int_{2}^{12}\frac{1}{x\sqrt{1+4x}}dx+4\int_{2}^{12}\frac{dx}{\sqrt{1+4x}}\right).$$
Now the second integral is pretty easy
$$2\int_{2}^{12}\frac{dx}{\sqrt{1+4x}}=\left[\sqrt{1+4x}\right]_{2}^{12}=4,$$
fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2540947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 4,
"answer_id": 3
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Help in solving linear recurrence relation I need to solve the following recurrence relation: $a_{n+2} + 2a_{n+1} + a_n = 1 + n$
My solution:
Associated homogeneous recurrence relation is:
$a_{n+2} + 2a_{n+1} + a_n = 0$
Characteristic equation: $r^2 + 2r + 1 = 0$
Solving the characteristic equation, we get: $r = -1$ w... | General technique uses generating functions. Define:
$\begin{equation*}
A(z) = \sum_{n \ge 0} a_n z^n
\end{equation*}$
Multiply the recurrence by $z^n$, sum over $n \ge 0$ and recognize the resulting sums:
$\begin{align*}
\sum_{n \ge 0} a_{n + 2} z^n
+ 2 \sum_{n \ge 0} a_{n + 1} z^n
+ \sum_{n \ge 0} a_n z^n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2541270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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If $p$ is prime, $p\ne3$ then $p^2+2$ is composite I'm trying to prove that if $p$ is prime,$p\ne3$ then $p^2+2$ is composite. Here's my attempt:
Every number $p$ can be put in the form $3k+r, 0\le r \lt 3$, with $k$ an integer. When $r=0$, the number is a multiple of 3, so that leaves us with the forms $3k+1$ and $3k+... | Yes, it is correct. However, there's a much shorter proof (which is essentially the same as yours): If $p$ is not divisible by $3$, then $p=3k+1$ or $p=3k+2$; then $p^2$ leaves a remainder of $1$ when divided by $3$ ($p^2\equiv 1\mod{3}$), so that $p^2+2$ is divisible by $3$.
Note by the way that you don't really need ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2543345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
Determine a probability of coin is fair in repeated trial The probability of getting head in a fair coin is 1/2 and the probability of getting tail in a fair coin is 1/2. While the probability of getting tail in a biased coin is 3/4 and probability of getting head in a biased coin is 1/4.
Suppose we don't know which on... | $$P(\text{8 heads in 10 tosses}|\text{fair coin})={10\choose{8}}\frac{1}{2}^{10}$$
$$P(\text{8 heads in 10 tosses}|\text{unfair coin})={10\choose{8}}\frac{1}{4}^{8}\frac{3}{4}^2$$
Thus, the probability that the coin was fair would be
$$\begin{align*}
P(\text{coin is fair})
&= \frac{P(\text{8 heads in 10 tosses}|\text{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2545479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to prove complex identity? I proved that $z^6+z^3+1 = (z^2-2z\cos(\frac{2π}{9})+1)(z^2-2z\cos(\frac{4π}{9})+1)(z^2-2z\cos(\frac{8π}{9})+1)$
Using this, how to show that $$2\cos(3 \theta)+1=8\left(\cos \theta-\cos\left(\frac{2π}{9}\right)\right)\left(\cos \theta-\cos\left(\frac{4π}{9}\right)\right)\left(\cos\theta-\... | Divide by $z^3$:
$$z^3+z^{-3}+1=(z+z^{-1}-2\cos(2\pi/9))(z+z^{-1}-2\cos(4\pi/9))(z+z^{-1}-2\cos(8\pi/9))$$
Put in $z=\exp(i\phi)$, noting that $z+z^{-1}=2\cos\phi$ and
$z^3+z^{-3}=2\cos3\phi$:
$$2\cos3\phi+1=(2\cos\phi-2\cos(2\pi/9))(2\cos\phi-2\cos(4\pi/9))(2\cos\phi-2\cos(8\pi/9)).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2546990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Prove that $\sum\limits_{r=0}^\infty\,\dfrac{(-1)^r\,n}{r!\,(r+1)!}\,\prod\limits_{k=1}^r\,\left(n^2-k^2\right)=(-1)^{n+1}$
If n is a positive integer, prove that:
$$\begin{align}n - {\frac{n(n^2-1^2)}{2!}}+{\frac{n(n^2-1^2)(n^2-2^2)}{2!3!}} + \ldots\phantom{aaaaaaaaaa} &&\\
+(-1)^r{\frac{n(n^2-1^2)(n^2-2^2)\cdots(n... | The product is:
$$\prod (n^2-k^2)=(n-r)(n-r+1)\cdots(n-1)(n+1)\cdots(n+r-1)(n+r)$$
$$=\frac{(2r+1)!}{n}{n+r \choose 2r+1}$$
The $n$ is because we are missing that term, the binomial symbol is because we recognize a product of $2r+1$ going down from $n+r$, and the factorial is because the binomial symbol is missing its ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2553579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Prove $5^n + 2 \cdot 3^n - 3$ is divisible by 8 $\forall n\in \mathbb{N}$ (using induction)
Prove $5^n + 2 \cdot 3^n - 3$ is divisible by 8 $\forall n\in \mathbb{N}$
Base case $n = 1\to 5 + 6 - 3 = 8 \to 8 \mid 8 $
Assume that for some $n \in \mathbb{N}\to 8 \mid 5^n + 2 \cdot 3^n - 3$
Showing $8 \mid 5^{n+1} + 2 ... | As $3^n$ is an odd number, $4\cdot 3^n\equiv 4~(mod~8)$, also $12\equiv 4~(mod~8)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2554949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 0
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Calculate the probability density function of two independent random variables: (R^3 + r^3)^(1/3) Random variables R and r are independent, both of them are uniform distributed and greater than zero. R distributes in (a,b), r in (c,d). I tried to solve the probability density function of (R^3 + r^3)^(1/3), but I couldn... | I recommend doing this one step at a time.
First let's find the CDFs and densities of $R^3$ and $r^3$:
$$F_{R^3}(x)=P(R^3 \leq x) = P(R \leq x^{1/3}) = \int_{-\infty}^{x^{1/3}}\frac{1}{b-a}1_{a \leq t^{1/3} \leq b} \ dt = \begin{cases}0 & x < a^3 \\ \frac{x^{1/3}-a}{b-a} & a^3\leq x\leq b^3 \\ 1 & x>b^3\end{cases}$$
s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2555580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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How to calculate this hard integral $\int_0^{\infty} \frac{\arctan(x)\sqrt{\sqrt{1+x^2}-1}}{x\sqrt{1+x^2}}\,dx$? How to prove that $\displaystyle \int_0^{\infty} \dfrac{\arctan(x)\sqrt{\sqrt{1+x^2}-1}}{x\sqrt{1+x^2}}\,dx= \frac{1}{4}\pi^2\sqrt{2}-\sqrt{2}\ln^2\left(1+\sqrt{2}\right)$ ?
It's very difficult and I have no... | Let $x = \tan u \implies dx = \sec^2u\ du$
$$\begin{align}\int_0^{\infty} \dfrac{\arctan(x)\sqrt{\sqrt{1+x^2}-1}}{x\sqrt{1+x^2}}\,dx =& \int_0^{\pi/2} \dfrac{\arctan(\tan u)\sqrt{\sec u-1}\sec u}{\tan u }\,du &\\=& \int_0^{\pi/2} \dfrac{u\sqrt{\sec u-1}}{\sin u }\,du&\\=& \int_0^{\pi/2} \dfrac{u\sqrt{1-\cos u}}{\sin u ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2557405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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Prove that $\sum_{n=0}^N\binom{2N-n}N2^n(n+1)=(1+2N)\binom{2N}N$ I used WolframAlpha to calculate a sum but it didn't show me the way :( Anybody has a hint or a solution for proving this sum?
$$\sum_{n=0}^N\binom{2N-n}N2^n(n+1)=(1+2N)\binom{2N}N$$
| Starting from
$$\sum_{n=0}^N {2N-n\choose N} 2^n (n+1)$$
we write
$$\sum_{n=0}^N {2N-n\choose N-n} 2^n (n+1)
= \sum_{n=0}^N 2^n (n+1) [z^{N-n}] (1+z)^{2N-n}
\\ = [z^N] \sum_{n=0}^N 2^n (n+1) z^n (1+z)^{2N-n}.$$
We may extend $n$ to infinity beyond $N$ because the sum term does not
contribute to the coefficient extracto... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2558601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Determinant of hermitian matrix Let $M=A+iB$ be a complex $n \times n$ Hermitian matrix. First of all we know that $$(\det M)^2=\det \begin{pmatrix} A & -B \\ B & A \end{pmatrix}.$$ Also $\det \begin{pmatrix} A & -B \\ B & A \end{pmatrix}$ is a polynomial in $n^2$ variables of degree $2n$. Is it true that $\det M$ is a... | The matrix is similar to
$$
S = T \hat{M} T^{-1}=
\begin{pmatrix} I & iI \\ I & -iI \end{pmatrix}
\begin{pmatrix} A & -B \\ B & A \end{pmatrix}
\begin{pmatrix} \frac12I & \frac12 I\\-\frac i2I& \frac i2I\end{pmatrix} =
\begin{pmatrix} A+Bi &\\ &A-Bi \end{pmatrix}=
\begin{pmatrix} M &\\ &M^* \end{pmatrix}=\begin{pmat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2559880",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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How to prove Big Theta on polynomial function? Im working on the Big-O notation and struggle to understand whether I have done enough to prove the following:
$5n^3+4n^2+4 \in \Theta (n^3)$
So based on the definition of $Θ(g(n)):$
Step $1$: $0 ≤ c_1 n^3 \leq 5n^3+4n^2+4 \leq c_2 n^3$
Divide the inequality by the larges... | Hint: $1 \le n^2 \le n^3$ for $n \ge 1\,$, so $5n^3 \;\le\; 5n^3+4n^2+4 \;\le\; 5n^3+4n^3+4n^3 = 13 n^3\,$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2560344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Parsing a question about the curl of a vector field on the surface of a sphere Given the following question and temptative answers:
I am not sure what the answer is for the $V(x,y)$ box.
Normally I would have converted the problem to spherical coordinates and solved it that way.
My guess is that when one is converting... | We have that $\nabla \times \mathbf{F}=\langle2yzx^3,-3y^2zx^2,-2\rangle$ and we have $z=-\sqrt{\frac{1-x^2-y^2}{6}}$ since $z\leq0$. Then we have that
$$\begin{align}
\iint_M(\nabla\times F)\cdot dS & = \iint_M \langle2yx^3z,-3y^2x^2z,-2\rangle \cdot\langle-z_{x},-z_{y},1\rangle\,dA\\
& = \iint_M\langle-2yx^3\sqrt{\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2562342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
joint pdf of two statistically independent random variables Let A and B be statistically independent, identically distributed (iid)
random variables having chi-square distribution with four degrees of freedom.
*
*calculate the joint pdf for $X=\frac{A-B}{A+B}$ and $Y=A+B$
*calculate the marginal pdfs of X and Y
S... | Thanks to @Henry I found out my mistakes, I used the wrong integration borders,
just for completeness i will answer my own question:
$f_X(x) = \int\limits_0^{\infty} \frac{1}{128}y^3(1-x^2)e^{-y/2}\, dy = \frac{3}{4} (1-x^2)$
$f_Y(y) = \int\limits_{-1}^{1} \frac{1}{128}y^3(1-x^2)e^{-y/2}\, dx = \frac{1}{96}y^3e^{-y/2}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2562448",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Solving the limit $\lim_{x\rightarrow+\infty}(\sqrt[5]{x^5-x^4}-x)$ Recently, I am struggling to solve the limit: $$\lim_{x\rightarrow+\infty}(\sqrt[5]{x^5-x^4}-x)$$
If I try to make some fraction with nominator $-x^4$ and some irrational denominator by multiplying, it becomes more complex. Can anyone help about this ... | Let $a=\sqrt[5]{x^5-x^4}$ and $b=x$. Then\begin{align}\sqrt[5]{x^5-x^4}-x&=a-b\\&=\frac{(a-b)(a^4+a^3b+a^2b^2+ab^3+b^4)}{a^4+a^3b+a^2b^2+ab^3+b^4}\\&=\frac{a^5-b^5}{a^4+a^3b+a^2b^2+ab^3+b^4}\\&=\frac{-x^4}{\sqrt[5]{x^5-x^4}^4+\sqrt[5]{x^5-x^4}^3\,x+\sqrt[5]{x^5-x^4}^2\,x^2+\sqrt[5]{x^5-x^4}\,x^3+x^4}\\&=\frac{-1}{\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2563344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 1
} |
Mathematical induction proof that $f(n)=\frac{1}{2}+\frac{3}{2}(-1)^n$
The function $f(n)$ for $n=0,1...$ has the recursive definition $$f(n)= \begin{cases} 2 & \text {for n=0} \\ -f(n-1)+1 & \text{for n=1,2...} \end{cases}$$
Prove by induction that the following equation holds: $$f(n)=\frac{1}{2}+\frac{3}{2}(-1)^n$... | $f(n)=1-f(n-1)$. We want to show that
$$f(n+1)=1-f(n)=1-[1-f(n-1)]=f(n-1)$$
$f(n+1)=\frac{1}{2} - \frac{3}{2}(-1)^n $
$f(n-1)= \frac{1}{2} + \frac{3}{2}(-1)^{n-1}$. Multiply this equality $-1$,then
$-f(n-1)= [\frac{1}{2} + \frac{3}{2}(-1)^{n-1}](-1)=-\frac{1}{2}+\frac{3}{2}(-1)^n$. Hence we obtain
$f(n-1)=\frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2564657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Proof by induction in trigonometry.
Prove that $\cos x +\cos 2x + \cos 3x + ...+ \cos nx =\cos \left(\dfrac{n+1}{2}x\right) \sin \left(\dfrac{nx}{2}\right)\csc \dfrac{x}{2}$
Attempt:
Clearly, $P(1)$ is true.
Assume $P(m)$ is true.
Thus, $P(m+1) = (\cos x +\cos 2x + \cos 3x + ...+ \cos mx)+ \cos((m+1)x)$
$= \cos \... | Formula to be used:
$\sin A- \sin B = \cos\left(\dfrac{A+B}{2}\right)\sin \left(\dfrac{A-B}{2}\right)$
Thus,
$\csc \left(\dfrac x 2 \right)\left(\cos \left(\dfrac{m+1}{2}x\right) \sin \left(\dfrac{mx}{2}\right)+ (\cos(m+1)x)\sin (\dfrac x 2)\right)$
$= \csc \left(\dfrac x 2 \right)\left(\dfrac 1 2 \left(\sin \dfrac{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2568173",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Transform $\iint_D \sin(x^2+y^2)~dA$ to polar coordinates and evaluate the polar integral.
Transform the given integral in Cartesian coordinates to polar coordinates and evaluate the polar integral:
(b) $\iint_D \sin(x^2+y^2)~dA$, where $D$ is the region in the first quadrant bounded by the lines $x=0$ and $y=\sqrt{3}... | No, it is not correct. The polar representation you give yields a quarter circle of radius $\sqrt{3\pi}$ on the first quadrant, which is obviously not the region described by the question.
The first thing I always do to solve these types of problems is to sketch the region of integration. In this case, it turns out to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2568843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Slope of The Tangent to $r=7\sin \theta$ , $\theta = \frac{\pi}{6}$ Find slope of tangent to $r=7\sin \theta$ , $\theta = \frac{\pi}{6}$
using $$\frac{dy}{dx} = \frac{\frac{dr}{dθ}\sin \theta + r \cos \theta}{\frac{dr}{dθ}\cos \theta-r \sin \theta}$$
I got
$$\frac {14\cos \theta \sin \theta}{7 \cos^2 \theta-7\sin \thet... | The curve has polar equation
$r=7 \sin t$
multiply both sides by $r$
$r^2=7r\sin t$
in rectangular coordinates becomes
$x^2+y^2=7y$
Equation of a circle having center $C(0,\;7/2)$ and radius $R=7/2$
The point of the circle when $t=\pi/6$ can be found intersecting the line, in polar form, $a:t=\pi/6$ with the given cir... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2571026",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
If $A+B = I_n$ and $A^2 +B^2 = O_n$ then $A$ and $B$ are invertible and $(A^{-1}+B^{-1})^n = 2^n I_n$ For $A$ and $B$ square matrices of size $n$, show that:
If $A+B = I_n$ and $A^2 +B^2 = O_n$ then $A$ and $B$ are invertible and $(A^{-1}+B^{-1})^n = 2^n I_n$
Where $I_n$ is the identity matrix of order $n$, and $O_n$ i... | First, let's show that A is invertible:
$$\begin{align} A+B=I\\B=I-A\\A^2+(I-A)^2=0\\I-2A+2A^2=0\\A(2(I-A))=I\\A^{-1}=2(I-A)\end{align}$$
Similarly, $B^{-1}=2(I-B)$, so $A^{-1}+B^{-1}=2(2I-A-B)=2I$ (because $A+B=I$), so:
$$(A^{-1}+B^{-1})^n=(2I)^n=2^nI$$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2573668",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
On the evaluation of $\int_0^\infty e^{-x^2} \sin \left (\frac{1}{4x^2} \right ) \, dx$ When attempting to find an alternative solution to this question here, which called for the evaluation of the integral
$$\int_0^\infty \frac{x \sin x}{1 + x^4} \, dx$$
using real methods, I ran up against the following improper inte... | Considering $$ I=\int e^{-x^2} \cos \left (\frac{1}{4 x^2} \right ) \, dx\qquad \qquad J=\int e^{-x^2} \sin \left (\frac{1}{4 x^2} \right ) \, dx$$
$$K=I+iJ=\int e^{-x^2+\frac{i}{4 x^2}}\,dx\qquad \qquad \qquad \qquad L=I-iJ=e^{-x^2-\frac{i}{4 x^2}}\,dx$$
$$-x^2+\frac{i}{4 x^2}=-\left(x^2-\frac{i}{4 x^2} \right)=-\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2585053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.