Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Prove that there are only 3 integer solutions to $2^a+3 ^b=5^c$ I have figured out that $a \pmod{2} = b\pmod{2} =c \pmod{2} $ by making a few modulo tables and using that both sides of the equation must be divisible by $5$, and I have found the three solutions, I'm just at a loss as to how to prove that those are the o... | There are three solutions which can all be found by elementary means.
If $b$ is odd
$$2^a+3\equiv 1 \bmod 4$$
Therefore $a=1$ and $b$ is odd.
If $b>1$, then $2\equiv 5^c \bmod 9$ and $c\equiv 5 \bmod 6$
Therefore $2+3^b\equiv 5^c\equiv3 \bmod 7$ and $b\equiv 0 \bmod 6$, a contradiction.
The only solution is $(a,b,... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Partial fraction expansion - handling multiple roots I am trying to solve the partial fraction expansion of the following:
$X(z) = \dfrac{3-15z}{9z^2-6z+1}$.
So I calculated the roots of $9z^2 - 6z + 1$.
This gives $n_1 = n_2 = \frac{1}{3}$
and $9z^2-6z+1 = 9(z-\frac{1}{3})(z-\frac{1}{3})$.
Now I don't know how to... | You could absolutely do it your way: \begin{align}
X(z)=\frac{3-15z}{9z^2-6z+1}&=\frac{3-15z}{9(z-\frac13)^2}\\
&=\frac19\bigg(\frac{3-15z}{(z-\frac13)^2}\bigg)\\
&=\frac19\bigg(\frac{A}{z-\frac13}+\frac{B}{(z-\frac13)^2}\bigg)
\end{align}
Not impossible, but not very nice.
Alternatively:
\begin{align}
X(z)=\frac{3-15... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3361504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Finding $f\in\mathbb{Q}[x]$ such that $f(\sqrt{2}+\sqrt{3})=\sqrt{2}$ and $\deg(f)\leq 3$. What's wrong with my approach?
I'd like to find a polynomial $f(x) \in \mathbb{Q}[x]$ satisfying
$$f(\sqrt{2}+\sqrt{3})=\sqrt{2}$$
and $\deg(f) \leq 3$.
What I've been trying is the following:
Since $f(\sqrt{2}+\sqrt{3})=\s... | The error appears to be somewhere in the unwritten details of "after comparing the coefficients of two polynomials, I found that...".
Here's a lowbrow approach:
Hint Denote $\beta := \sqrt{2} + \sqrt{3}$. We're looking to solve $\sum_{i = 0}^3 a_i \beta^i = \sqrt{2}$ in rational numbers $a_i$. Computing gives that
$$\b... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Use $T(x,y)=\begin{pmatrix}\frac{\sin x}{\cos y}\\ \frac{\sin y}{\cos x}\end{pmatrix}$ to evaluate $\sum_{n=1}^{\infty}\frac{1}{n^2}$ Let $O=\{(x,y) \in \mathbb R^2: x,y>0, x+y<\frac{\pi}{2}\}$ and $P=(0,1)\times (0,1)$.
Then $T:O \to \mathbb R^2, T(x,y)=\begin{pmatrix}\frac{\sin x}{\cos y}\\ \frac{\sin y}{\cos x}\end{... | Well,
$$ \frac{1}{(n+1)^2}=\iint_{(0,1)^2}(xy)^n\,dx\,dy $$
implies
$$ \sum_{n\geq 0}\frac{1}{(2n+1)^2}=\iint_{(0,1)^2}\frac{dx\,dy}{1-x^2 y^2}.$$
Since the diffeomorphism $T$ maps the triangle $O=\{(a,b):a,b>0,a+b<\frac{\pi}{2}\}$ into the square $P=(0,1)^2$, by letting $x=\frac{\sin a}{\cos b},y=\frac{\sin b}{\cos a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3364583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Limit of $f(x, y)$ at $(0, 0)$ I need to show
$$\lim_{(x, y) \rightarrow (0,0)} \frac{xy(x^2-y^2)}{(x^2+y^2)^{3/2}} = 0
$$
Not really sure how to go about this without using the epsilon delta definition, which I would prefer not to. Any sort of help is appreciated.
Edit: I do have the inequality:
$$\frac{xy(x^2-y^2)}{... | Let $(x,y) \not =(0,0)$.
$0\le |\dfrac{xy(x^2-y^2)}{(x^2+y^2)^{3/2}}| \le \dfrac{|xy||x^2+y^2|}{(x^2+y^2)^{3/2}} $
$\le \dfrac{(x^2+y^2)^2}{(x^2+y^2)^{3/2}}= (x^2+y^2)^{1/2}.$
Used:
1)$|x^2-y^2| \le |x^2+y^2|;$
2)$x^2+y^2 \ge |xy|$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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a) Find a base for $W_1 \cap W_2$ and b) Find a base for $W_1 + W_2$ Let
$$W_1 = \left\{ \begin{pmatrix}
a & b & c \\
-b & a & b \\
c & b & a \\
\end{pmatrix} :\, a, b, c \in \Bbb R \right\}$$
$$W_2 = \left\{
\begin{pmatrix}
a & 0 & 0 \\
b & 0 & c \\
c & b & a \\
\end{pmatrix} :\, a, b, c \i... | a) Since $W_1\cap W_2=\{0\}$, the only basis of $W_1\cap W_2$ is the empty set.
b) Since $W_1\cap W_2=\{0\}$, a way of finding a basis $B$ of $W_1+W_2$ is to take a basis $B_1$ of $W_1$ and a basis $B_2$ of $W_2$ and then to take $B=B_1\cup B_2$. It is natural to take$$B_1=\left\{\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3365840",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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A First Course in Mathematical Analysis, Ch 1, 1.6 Exercises, Section 1.3, ex 2 Prove that
\begin{align*}
(a^{2}+b^{2})(c^{2}+d^{2}) & \ge(ac+ad)^{2}\\
a^{2}c^{2}+a^{2}d^{2}+b^{2}c^{2}+b^{2}d^{2} & \ge a^{2}c^{2}+2a^{2}cd+a^{2}d^{2}\\
b^{2}c^{2}+b^{2}d^{2} & \ge 2a^{2}cd\\
b^{2}c^{2}-2a^{2}cd+b^{2}d^{2} & \ge 0\\
b^{2}... | As written, the inequality is false.
$$(ac+ad)^2=a(c+d)^2=a^2c^2+2a^2cd+a^2d^2.$$
If $b=0 \text{ but } acd \neq 0 \text{ with } cd \gt 0$, then
$$(a^2+b^2)(c^2+d^2) = a^2(c^2+d^2) = a^2c^2+a^2d^2 \lt a^2c^2+2a^2cd+a^2d^2.$$
As noted in the comments, I suspect the right-hand side is supposed to be $(ac+\color{red} b d)^... | {
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"url": "https://math.stackexchange.com/questions/3367344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Rational $a$ such that $\sin(a\pi)$ or $\tan(a\pi)$ have the form $\pm\sqrt{a_1}\pm\sqrt{a_2}\pm\cdots\pm\sqrt{a_n}$ for rational $a_i$ I'm interested in simple values of trigonometric functions, e.g., $\sin 30^{\circ} = 1/2$ or $\sin 18^{\circ} = \frac{\sqrt{5}-1}{4}$.
Let $S$ be the set of real numbers that can be w... | Most probably this is the complete list (between 0 and 90 degrees):
$$\begin{array}{|c|c|c|} \hline
&\sin &\tan\\ \hline
0^{\circ} &0 &0\\ \hline
7.5^{\circ} & &(\sqrt{2}-1)(\sqrt{3}-\sqrt{2})\\ \hline
15^{\circ} &(\sqrt{6}-\sqrt{2})/4 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3370069",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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how to solve $\lim_{n\to\infty}{\left(\sum_{k=1}^{n}{\frac{1}{\sqrt{n^2+k}}}\right)^{n}}$? $\displaystyle\left(\sum_{k=1}^{n} \frac{1}{\sqrt{n^{2}+1}}\right)^{n}\ge\left(\sum_{k=1}^{n} \frac{1}{\sqrt{n^{2}+k}}\right)^{n}\ge\left(\sum_{k=1}^{n} \frac{1}{\sqrt{n^{2}+n}}\right)^{n}$
left=$\displaystyle\lim_{n\to\infty}{\l... | Another solution: Let $$S_n= \sum_{k=1}^{n} \frac{1}{n}\frac{1}{\sqrt{1+\frac{k}{n^2}}}$$
We will prove that $$ \frac{2x+2}{2+x}<\sqrt{1+x} < 1+\frac{x}{2}, x>0~~~~(1)$$
Right one: is nothing but $$2\sqrt{1+x}=-1-(1+x)-[1-\sqrt{1+x}]^2 \le 0.$$
The left one isnothing but $$\frac{2(1+x)}{1+1+x} < \sqrt{1+x} \Rightar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3370438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 1
} |
Different proof for the inequality $b \gcd(a,c)^2+ a \gcd(b,c)^2-c \gcd(a,b)^2 \le abc$? In Encyclopedia of Distances, there is given a distance on natural numbers without zero:
$$d(a,b) = \log\left( \frac{ab}{\gcd(a,b)^2}\right)\,.$$
Taking (again for a reference see the Encyclopedia of Distances), the Schoenberg tran... | Write
$$abc-b\gcd(a,c)^2-a\gcd(b,c)^2=c\left(a-\frac{\gcd(a,c)^2}{c}\right)\left(b-\frac{\gcd(b,c)^2}{c}\right)-\frac{\gcd(a,c)^2\gcd(b,c)^2}{c}.$$
Since $ac\ge\gcd(a,c)^2$ and $bc\ge\gcd(b,c)^2$, we see that
$$abc-b\gcd(a,c)^2-a\gcd(b,c)^2\ge-\frac{\gcd(a,c)^2\gcd(b,c)^2}{c}.$$
We only need to show that
$$\frac{\gcd(a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3371958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Solve the equation $\sqrt{a(2^{x}-2)+1}=1-2^{x}$ for every value of the parameter $a$. Question : Solve the equation $\sqrt{a(2^{x}-2)+1}=1-2^{x}$ for every value of the parameter a.
I have solved the problem as follows
$\sqrt{a\left(2^{x}-2\right)+1}=1-2^{x}$
$a\left(2^{x}-2\right)+1=\left(1-2^{x}\right)^{2}=2^{2 x}... | With $z:=2^x-2$, $$\sqrt{az+1}=-z-1$$
or after rewriting,
$$z(z+2-a)=0,z\le-1.$$
Hence
$$z=a-2\le-1,$$ which is $$2^x=a\le1$$
or
$$x=\log_2a,\\0<a\le1.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3376613",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Interesting function : $f(x)=\frac{\ln (\frac{x+2}{x+1})}{\ln (1+\frac{1}{x})}$ , $x>1$ Question :
Prove this function :
$$f(x)=\dfrac{\ln \left(\dfrac{x+2}{x+1}\right)}{\ln \left(1+\dfrac{1}{x}\right)},\quad x>1$$
is increasing.
I don't know how I solve by my effort is :
Derivative of $f$ is :
$$f'(x)=\dfrac{(x+... | EDIT
Here's a proof without expansions.
We wish to prove that
$$f(x) = \frac{\log (\frac{x+2}{x+1})}{\log (\frac{x+1}{x})}$$
is an increasing function, i.e. that
$$f'(x) = \frac{(x+2) \log \left(\frac{1}{x+1}+1\right)-x \log \left(\frac{1}{x}+1\right)}{x (x+1) (x+2) \log ^2\left(\frac{1}{x}+1\right)}>0$$
Since the de... | {
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"url": "https://math.stackexchange.com/questions/3379519",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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How to find number of distinct terms in a multinomial expansion?
Find the number of distinct terms in the expansion of
$$\left(x+\frac{1}{x}+\frac{1}{x^2}+x^2\right)^{15}$$
(with respect to powers of $x$)
I saw that the formula for the number of distinct terms (or dissimilar) in a multinomial expansion $(x_1+x_2+x_3+... | Let us equivalently find the number of different monomials in
$$
f=(x^2)^{15}\left(\frac{1}{x^2}+\frac{1}{x}+x+x^2\right)^{15}
=
\left(1 + x + x^3 + x^4\right)^{15}
$$
(after collecting them and writing the polynomial w.r.t. the basis $1,x,x^2,\dots$), which is a reciprocal polynomial of degree $4\cdot 15=60$. It is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3380253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Ramanujan's Nested Radical By noting Ramanujan's Nested Radical, we have
$3 = \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}$
On the other hand, we can manipulate the number $4$ by applying the similar principle. Here we have
$\begin{aligned} 4 & = \sqrt{16} \\ & = \sqrt{1+15} \\ & = \sqrt{1+2 \cdot \dfrac{15}{2}} \\ & ... | By Ramanujan's nested radical, we can also get this curious identity;
$$n+m=\sqrt{m^2+n\sqrt{m^2+(n+m)\sqrt{m^2+(n+2m)\sqrt{..}}}}$$
Ramanujan proved this identity, and if you want a look at how it's done in a simple manner (not-rigorous), check out this blog. And by plugging in $4$, we get;
$$3+1=\sqrt{1+3\sqrt{1+4\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3382458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Solution of a limit of a sequence $\frac{\sqrt{4n^2+1}-2n}{\sqrt{n^2-1}-n}$ I'm trying to solve the limit of this sequence without the use an upper bound o asymptotic methods:
$$\lim_{n\longrightarrow\infty}\frac{\sqrt{4n^2+1}-2n}{\sqrt{n^2-1}-n}=\left(\frac{\infty-\infty}{\infty-\infty}\right)$$
Here there are my diff... | Note
\begin{eqnarray}
\lim_{n\to\infty}\frac{\sqrt{4n^2+1}-2n}{\sqrt{n^2-1}-n}&=&\lim_{n\to\infty}\frac{(\sqrt{4n^2+1}-2n)(\sqrt{4n^2+1}+2n)}{(\sqrt{n^2-1}-n)(\sqrt{n^2-1}+n)}\cdot\frac{\sqrt{n^2-1}+n}{\sqrt{4n^2+1}+2n}\\
&=&-\lim_{n\to\infty}\frac{\sqrt{n^2-1}+n}{\sqrt{4n^2+1}+2n}\\
&=&-\lim_{n\to\infty}\frac{\sqrt{1-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3384280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Substitution integral with $t=x^3$
$$
\int \frac{3}{t(3+\sqrt[3]{t})} \mathrm{d} t . \text { Substitution } t=x^{3}
$$
$$
\int \frac{3}{x^3(3+\sqrt[3]{x^3})}3x^2
$$
$$
\int \frac{9}{x(3+{x})}
$$
$$
9\int \frac{1}{3x+{x^2}}
$$
Now i'm stuck
my calculation so far
| You could do partial fraction decomposition, but instead let's multiply the top and bottom by $\frac{1}{x^2}$:
$$3\int\frac{3}{3x+x^2}dx = 3\int \frac{\frac{3}{x^2}}{\frac{3}{x}+1}dx$$
Notice that the numerator is the derivative of the denominator times $-1$, so we have an integral of the form $\int\frac{du}{u}$, which... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3386227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the roots of $z^4-3z^2+1=0$ in polar form. Question :
Prove that the solutions of $z^4-3z^2+1=0$ are given by :
$$z=2\cos{36^\circ},2\cos{72^\circ},2\cos{216^\circ},2\cos{252^\circ}$$
My work :
First of all, i want ro find the roots with quadratic formula
$\begin{align}
&(z^2)^2-3z^2+1=0\\
&z^2=\dfrac{3\pm \sqrt{5... | Note that
$$\cos36 = -\cos216 = \frac{1+\sqrt5}{4}$$
$$\cos72 = -\cos252 = \frac{1-\sqrt5}{4}$$
Then,
$$(x-2\cos36)(x-2\cos216) = x^2 - \frac{3+\sqrt5}{2}=0$$
$$(x-2\cos72)(x-2\cos252) = x^2 - \frac{3-\sqrt5}{2}=0$$
Together,
$$(x-2\cos36)(x-2\cos216)(x-2\cos72)(x-2\cos252)=x^4-3z^2+1=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3386733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
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Domain of inequation inside squared root How should I go when restricting the roots of the inequation:
$\sqrt {x^2+5x+6} - \sqrt {x^2-x+1} \lt 1$?
By restricting both the squared roots, I know that:
$x \le 3$ and $x \ge -2$
However when simplifying the whole inequation, I get the two roots:
$\frac{-13-\sqrt{73}}{16}$ a... | As $x^2-x+1 > 0$ it has no effect on the domain. However, $x^2+5x+6$ is negative in the interval $(-3,-2)$. So we must keep this in mind when we are crafting a solution
Consider $$\sqrt{(x+2)(x+3)} < 1+\sqrt{x^2-x+1}$$
$$x^2+5x+6 < x^2-x+2 + 2\sqrt{x^2-x+1}$$
$$3x+2 < \sqrt{x^2-x+1}$$
$3x+2$ being linear and $\sqrt{x^2... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $\alpha$, $\beta$, $\gamma$ and $\delta$ be the roots of the polynomial $x^4+px^3+qx^2+rx+s$, prove the following.
If $\alpha$, $\beta$, $\gamma$ and $\delta$ be the roots of the polynomial $x^4+px^3+qx^2+rx+s$, show that $$(1+\alpha^2)(1+\beta^2)(1+\gamma^2)(1+\delta^2)=(1-q+s)^2+(p-r)^2.$$
I have tried putting
\... | Another way using Transformation of equation:
We have $$x^4+qx^2+s=-x(px^2+r)$$
Squaring both sides
$$(x^2)^4+q^2(x^2)+s^2+2q(x^2)^3+2qs(x^2)+2s(x^2)^2=x^2(p^2(x^2)^2+r^2+2pr(x^2))$$
Replacing $x^2+1=y$
$$(y-1)^4+q^2(y-1)^2+s^2+2q(y-1)^3+2qs(y-1)+2s(y-1)^2=(y-1)(p^2(y-1)^2+r^2+2pr(y-1))$$
$$\iff y^4+\cdots+1+q^2+s^2-2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3390796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
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How many different ways to select a subset
We want to select three subsets A, B, and C of {1, 2, . . . , n} so that A ⊆ C,
B ⊆ C, and A ∩ B $\ne$ ∅. In how many different ways can you do this?
This is my attempt:
Let A= $A_1$, B= $A_2$ and C = $A_1 \cup A_2$. The set = $A_1 \cup A_2 \cup A_3$. Then $3^n$ ways?
| We want to ensure that $A \subset C$ and $B \subset C$ such that $A \cap B \neq \emptyset$.
We'll suppose that $A \cap B = \emptyset$, which are the cases that we don't want to account... Having in hands that formula, we can just subtract from all possible cases of sets $A$ and $B$ without restriction.
So suppose that... | {
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"url": "https://math.stackexchange.com/questions/3394240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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If $f(x)=x^2+ax+b$ divides $f(x+2)f(x-2)$, find the minimum value of $f$. I am stuck with this Precalculus problem about polynomial functions. The problem:
Consider $f(x)=x^2+ax+b$ with $a^2-4b>0$. Let $\alpha$ and $\beta$ be the roots of $f$. Assume that $f(x)$ divides $f(x+2)f(x-2)$. Then
*
*Show that $\alpha=\bet... | $ f(x) = ( x - \alpha) ( x - \beta) = (x - c + 1)( x - c - 1) = (x-c)^2 - 1$
Hence, the minimum value is -1.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3394378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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Taylor series of $\sinh{(x)}$ at $\ln{(2)}$. Determine the Taylor series of $\sinh{(x)}$ about $x = \ln{(2)}$.
Equating each derivative at $x = \ln{(2)}$ gives:
\begin{equation*}
\begin{split}
f'(\ln{(2)}) &= \frac{2+\frac{1}{2}}{2} = \frac{5}{4} \\
f''(\ln{(2)}) &= \frac{2-\frac{1}{2}}{2} = \frac{3}{4} \\
f^{(3)}(\l... | $$\sinh (\ln 2)= \sum_{k=0}^{\infty} \left(\frac{2-(-1)^k\frac{1}{2}}{k!}\right) (x-\ln 2)^k$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3395794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculating Legendre symbol : $\left(\frac{3}{2^{2^n}+1}\right)$ , n being positive. I want to find the Legendre symbol : $\left(\frac{3}{2^{2^n}+1}\right)$
for any positive n.
Also I want to find the Legendre symbol for: $\left(\frac{5}{2^{2^n}+1}\right)$, when $n>1$
Solution:
Using Gaussian reciprocity law: $\left(... | Wikipedia's Legendre symbol page says the definition is
$$\left({\frac {a}{p}}\right)={\begin{cases}1&{\text{if }}a{\text{ is a quadratic residue modulo }}p{\text{ and }}a\not \equiv 0{\pmod {p}},\\-1&{\text{if }}a{\text{ is a non-quadratic residue modulo }}p,\\0&{\text{if }}a\equiv 0{\pmod {p}}.\end{cases}} \tag{1}\la... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3397034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find $x$ that minimizes $\max_{1 \leq i \leq 3} f_i(x)$ My question is:
For $0 < a_1 \leq a_2 \leq a_3$ fixed, find $x>0$ that minimizes
\begin{align*}
g(x) = \max_{1\leq i \leq 3} f_i(x).
\end{align*}
where
\begin{align*}
f_i(x) = \left(\frac{a_i - x}{a_i + x}\right)^2
\end{align*}
I do not know how to start. A... | I think I have a solution to my question.
Answer: the minimizer is $x = \sqrt{a_1 a_3}$.
Let $0 < \underline{a} < \overline{a}$ be fixed. We study the relative position of $f_{\underline{a}}$ and $f_{\overline{a}}$ defined by
\begin{align*}
f_{\underline{a}}(x) = \left(\frac{\underline{a} - x}{\underline{a} + x}\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3399153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Which is bigger 5^(√3) and 4^(√5)? How do you know which one is bigger between $5^\sqrt{3}$ and $4^\sqrt{5}$?
For my method I used in $2^\sqrt{3}$ and $3^\sqrt{2}$ I put both numbers in the function $f(x)=x^\sqrt{3}$ so $f(2^\sqrt{3})$ become $2^3$ which is equal to 8 and $f(3^\sqrt{2})$ become $3^\sqrt{6}$ and $3^\sqr... | Take both to the power of $\sqrt{5}$
$\left(5^\sqrt{3}\right)^\sqrt{5}=5^{\sqrt{15}}<5^{\sqrt{16}}=5^4=625$
$\left(4^\sqrt{5}\right)^\sqrt{5}=4^5=1024$
so we have that $4^\sqrt{5}>5^\sqrt{3}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3399681",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
a line tangent to curve $y = \frac{x}{2−2x}$ goes through (1, -1) $y = \frac{x}{2−2x}$
$y = \frac{f(x)}{g(x)} \rightarrow y' = \frac{f'(x) \cdot g(x) - g'(x)\cdot f(x)}{{g(x)}^2}$
$y = \frac{x}{2−2x}$
$y' = \frac{(2-2x)-(x \cdot (-2))}{{(2−2x)}^2}$ $\rightarrow$ $y' = \frac{(2-2x+2x)}{{(2−2x)}^2} = \frac{2}{{(2-2x)}^2}... | Your derivative is absolutely correct.
Equation of a line is $(y-y_1)=m(x-x_1)$ where $(x_1,y_1) $ is a point that lies on the line
Here $m = \frac{2}{(2-2x)^2}$
Now we know that $(1,-1)$ lies on the line and the gradient of the line is $\frac{2}{(2-2k)^2}$
Substitute everything
We get $y=\frac{2}{(2-2k)^2}(x-1)-1$
Now... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3399947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
If $A$ is a rotation matrix, then $||Ax||=||x||$. Attempt:
Let $$
A =
\begin{bmatrix}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{bmatrix},
x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}
$$
Then,
$$Ax =
\begin{bmatrix}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{bmatrix} \cdot \begin{bmatri... | What you have done is correct. The reason for putting the condition that $0\leq \theta \leq \pi$ is that the 'rotation of the plane' plays the role here. To make reader understand better, they have put the condition. For example , Let $\theta = 359^o$ then one might get confused whether rotation is by $1^o$ or $359^0$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3401051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
help solving for delta in epsilon delta proof $$\lim_{x\to 9} \frac{1}{\sqrt{x}} = \frac{1}{3} $$
so the two statements are
$$0<|x-9|<\delta$$
$$\left|\frac{1}{\sqrt{x}}-\frac{1}{3}\right|<\epsilon$$
I've tried to multiply by the conjegate to get
$$\frac{\frac{1}{x}-\frac{1}{9}}{\sqrt{\frac{1}{x}}+\frac{1}{3}}<\epsilo... | Note that$$\left\lvert\frac1{\sqrt x}-\frac13\right\rvert=\frac{\left\lvert3-\sqrt x\right\rvert}{3\sqrt x}=\frac{\lvert9-x\rvert}{3\sqrt x\left(3+\sqrt x\right)}.$$Now, if $\lvert x-9\rvert<5$, then $x>4$ and therefore $\sqrt x>2$. Also, $3+\sqrt x>5$. So $3\sqrt x\left(3+\sqrt x\right)>30$. Therefore, if you take $\d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3403270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Diophantine equations with cubic roots.
Find all positive, integral $x, y$ such that
$$x^3-y^3=xy+61$$
Little work:
$$(x-y)(x^2+xy+y^2)-xy=61$$
Also $x^3-xy-y^3$ seems to to be unable to be "completed by the square," or factored, so I'm stuck. Thanks!
| Draw some pictures. There is an integral point on it $(6,5)$
As soon as real $x > 6,$ we get
$$ x-1 < y < x $$
so that $y$ cannot be an integer when $x > 6$
Tuesday:
We know that $x > y$ are positive. So,
$(x-y)^2 > 0$ or $x^2 - 2xy + y^2 > 0,$ or $x^2 + y^2 > 2xy.$ Add $xy$ to both sides with positive $x,y$ we get
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3403526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
$\int \frac{\sin x}{1+2\sin x}dx$ calculate:
$$\int \frac{\sin x}{1+2\sin x}dx$$
I tried using $\sin x=\dfrac{2u}{u^2+1}$, $u=\tan \dfrac{x}2$ and after Simplification:
$$\int \frac{2u}{u^2+4u+1}×\frac{2}{u^2+1}du$$
and I am not able to calulate that.
| Simplify first before substituting,
$$\int \frac{\sin x}{1+2\sin x}dx
=\frac12\int dx -\frac12 \int \frac{1}{1+2\sin x}dx$$
$$=\frac12 x -\frac12 \int \frac{1+\tan^2\frac x2}{\tan^2\frac x2 +4\tan\frac x2 +1}dx$$
$$=\frac12 x -\int \frac{d(\tan \frac x2)}{\left(\tan^2\frac x2 +2\right)^2 -3}$$
$$=\frac12 x +\frac{1}{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3404771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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How many ways from point A to point B? I am trying to solve this problem
In how many ways can we go from point $A$ to point $B$ if we are only allowed to move along the arrows?
I use combinations and got an answer of $20$.
Since this is 3D grid
The height is $3$ units, length is $1$ unit and width is $1$ unit. Thus we... | Graphical method:
We can easily calculate the number of different paths from $A$ to $B$ by adding the numbers of related sub-paths in the graphic.
Algebraic method:
We introduce a coordinate system and look for the number of different paths to go from $A=(0,0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3406911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A bunch of lilac A bunch of lilac consists of flowers with $4$ or $5$ petals. The number of flowers and the total number of petals are perfect squares. Can the number of flowers with $4$ petals be divisible by the number of flowers with $5$ petals?
What I produced:
Total Flowers = $x + y$
Petal Total $5x + 4y$
$x | y$ ... | You are correct in the first steps.
Continuing your argument: $x = n^2 - 4m^2$ and $y = 5m^2 - n^2$.
Hence we have $5m^2 - n^2 = k(n^2 - 4m^2)$. This can be rewritten as $\frac{4k + 5}{k + 1} = \frac{n^2}{m^2}$.
Now notice that, since $k$ is integer, we have $\gcd(4k + 5, k + 1) = \gcd(1, k + 1) = 1$, hence both $4k + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3407213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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formula for $\left(1\cdot2\cdot...\cdot k\right)+...+\left(n\left(n+1\right)...\left(n+k- 1\right)\right)$ proof if $k$ is a constant then for every $n$ natural numbers we have:
$$\left(1\cdot2\cdot...\cdot
k\right)+\left(2\cdot3\cdot...\cdot\left(k+1\right)\right)+...+\left(n\left(n+1\right)...\left(n+k-
1\right)\rig... | Note that by the Hockey-stick identity we have that
$$1\cdot2\cdots k+2\cdot3\cdots(k+1)+\dots+n\cdot(n+1)\cdots(n+k-1)$$
$$\begin{align}
&=k!\binom{k}{k}+k!\binom{k+1}{k}+\dots+k!\binom{n+k-1}{k}\\
&=k!\sum_{i=k}^{n+k-1}\binom{i}{k}\\
&=k!\binom{n+k}{k+1}\\
&=k!\cdot\frac{n\cdot(n+1)\cdots(n+k)}{(k+1)!}\\
&=\frac{n\cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3407338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Find $\lim_{x \to \ 0}{\frac{1-\sqrt{\cos x}}{1-\cos\sqrt{x}}}$ without using the l'Hospital rule The task is to evaluate
$$\lim_{x \to \ 0}{\frac{1-\sqrt{\cos x}}{1-\cos \sqrt{x}}}.$$
I tried to use some trigonometric identities such as
$$\lim_{x \to \ 0}{\frac{1-\sqrt{\cos\left(x\right)}}{1-\cos\left(\sqrt{x}\right)... | Using the standard Taylor expansion of $\cos(x)$, you should have
$${\frac{1-\sqrt{\cos\left(x\right)}}{1-\cos\left(\sqrt{x}\right)}}=\frac{\frac{x^2}{4}+O\left(x^4\right) } {\frac{x}{2}+O\left(x^2\right) }=\frac{x}{2}+O\left(x^2\right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3408177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 0
} |
Show that $\frac{3\> + \>\cos x}{\sin x}$ cannot have any value between $-2\sqrt2$ and $2\sqrt2$ Show that
$$\dfrac{3+\cos x}{\sin x}\quad \forall \quad x\in R $$
cannot have any value between $-2\sqrt{2}$ and $2\sqrt{2}$.
My attempt is as follows:
There can be four cases, either $x$ lies in the first quadrant, secon... | Use the half-angle expressions
$\cos x = \frac{1-\tan^2\frac x2}{1+\tan^2\frac x2}$ and
$\sin x = \frac{2\tan\frac x2}{1+\tan^2\frac x2}$ to express
$$I=\frac{3+\cos x}{\sin x}=
\frac{2}{\tan\frac x2} +\tan \frac x2$$
Note
$$I^2=\left(\frac{2}{\tan \frac x2} +\tan \frac x2\right)^2
=\left(\frac{2}{\tan \frac x2} -\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3411788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
} |
A simpler way to evaluate $\int {dx\over \sin^2x\cos^4x}$
Evaluate:
$$
\int {dx\over \sin^2x\cos^4x}
$$
I've started by using identities:
$$
{1\over \sin^2x} = 1+\cot^2x\\
{1\over \cos^2x} = 1+\tan^2x
$$
So the integral becomes:
$$
\begin{align}
I &= \int (1+\cot^2x)(1+\tan^2x)^2dx \\
&=\int (1+\cot^2x)(1+2\tan^2x ... | $I = \displaystyle\int \dfrac{1}{\sin^2{x}\cos^4{x}} dx = \displaystyle\int \sec^2{x}\dfrac{\left(\tan^2{x}+1\right)^2}{\tan^2{x}} dx$. Now substitute $y = \tan{x}$.
$I= \displaystyle\int \dfrac{(y^2+1)^2}{y^2} dy = \frac{y^3}{3}+2y-\frac{1}{y} +C$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3414034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the smallest integer $n$ such that $5x^5 + (\log x)^{5^5} \in O(x^n).$ Find the smallest integer $n$ such that $5x^5 + (\log_2 x)^{5^5}$ is $O(x^n).$
I know that $5x^5$ is $O(x^5),$ and that $\log_2 x$ is $O(x).$ Since $5^5=3,125$ and $\log_2 x$ is $O(x),$ I know that I can use a Theorem to get
$$
\begin{aligned}
... | The following shows that $(\log x)^{5^5} \in O(x)$: For positive $y$ we have
$$
\begin{aligned}
y &< 2^y &&\text{can prove by induction}\\
\sqrt[5^5]{x} &< 2^{\sqrt[5^5]{x}} &&\text{set }y = \sqrt[5^5]{x}\text{ for positive } x\\
\log_2\left(\sqrt[5^5]{x}\right) &<\log_2\left(2^{\sqrt[5^5]{x}}\right) &&\text{take $\log... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3414382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Divisibility by 7 of a number consisting of 0 and 1's A decimal number is of arbitrary length consisting of 0 and 1 only i.e. (10,111,10000,11111,1010, number of digits in the number can be upto 100 )
Can this number ever be divisible by 7 if yes, is there any efficient way to list all those numbers
| $10 \equiv 3 \pmod 7$ and so $10^2 \equiv 2\pmod 7$ and $10^3 \equiv -1\pmod 7$ and $10^4 \equiv -3 \pmod 7$ and $10^5 \equiv -2 \pmod 7$ and $10^6 \equiv 1 \pmod 7$.
Let $N = 10^{a_n} + 10^{a_{n-1}} + ....... + 10^{a_0}$ where no $a_i = a_j$ if $i\ne j$ be a number consisting of only $0$s and $1$.
Then $N \equiv b_n +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3417330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Given a point and a line in homogeneous coordinates, what is the coordinate of the projection of the point on the line? Knowing $M$ and $L$ in homogeneous coordinates, how can we determine the coordinates of the point $P$?
| it still depends on what kind of projection is required; Given the constraints in the original post, it looks like an orthographic parallel projection. Both central projection and parallel projection (oblique or orthographic) are computation-friendly redefined as a common algebraic formulation equation 3.1 in Household... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3419542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Solve : $\sin\left(\frac{\sqrt{x}}{2}\right)+\cos\left(\frac{\sqrt{x}}{2}\right)=\sqrt{2}\sin\sqrt{x}$ Solve : $\sin\left(\dfrac{\sqrt{x}}{2}\right)+\cos\left(\dfrac{\sqrt{x}}{2}\right)=\sqrt{2}\sin\sqrt{x}$
$$\dfrac{1}{\sqrt{2}}\cdot\sin\left(\dfrac{\sqrt{x}}{2}\right)+\dfrac{1}{\sqrt{2}}\cos\left(\dfrac{\sqrt{x}}{2}\... | First,
$\sin \alpha = \sin \beta \iff
\beta \equiv \alpha \pmod{2 \pi}
\quad \text{or} \quad
\beta \equiv \pi - \alpha \pmod{2 \pi}$
So, from
$$ \sin\left(\dfrac{\sqrt{x}}{2}+\dfrac{\pi}{4}\right) = \sin\sqrt{x} $$
You could have argued
$$ \sqrt x = 2m\pi + \dfrac{\sqrt{x}}{2}+\dfrac{\pi}{4}
\qquad \text{or} \q... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3420908",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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A general formula to generate functions of power series I have been playing with Maclaurin series lately, I have been able to come across this:
$\dfrac{1}{1+x}=1-x+x^2-x^3+x^4-x^5...$
$\dfrac{1}{(1+x)^2}=1-2x+3x^2-4x^3+5x^4-6x^5+7x^6...$
I found out by accident that:
$\dfrac{1-x}{(1+x)^3}=1-2^2x+3^2x^2-4^2x^3+5^2x^4+6^... | The most efficient way to obtain such formulae is to compute the Newton series for $k$-th powers, and then use the fact that $1/(1-x)^{k+1} = \sum_{n=0}^∞ \binom{n+k}{k} x^n$ for $|x| < 1$, which is easy to prove by induction (or by observing that the coefficients for the series are a column of Pascal's triangle).
For ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3422095",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Find the sum of series: $\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{97}+\sqrt{98}}+\frac{1}{\sqrt{99}+\sqrt{100}}$ Find the sum of series:
$$\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{5}+\sqrt{6}}+...+\frac{1}{\sqrt{97}+\sqrt{98}}+\frac{1}{\sqrt{99}+\sqrt{100}... | Let $S_m$ denote $\sum_{j=1}^m\sqrt{j}$. (In terms of Hurwitz eta functions $S_n=\zeta(-\frac12)-\zeta(-\frac12,\,m+1)$.) Take $n=50$ in $$\sum_{k=1}^n\frac{1}{\sqrt{2k-1}+\sqrt{2k}}=\sum_{k=1}^n(\sqrt{2k}-\sqrt{2k-1})=\sqrt{2}\sum_{k=1}^n\sqrt{k}-\sum_{2|k,\,1\le k\le 2n}\sqrt{k}\\=2\sqrt{2}\sum_{k=1}^n\sqrt{k}-\sum_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3422567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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How do I eliminate the repeating cases? Let $K$ be the field with exactly $7$ elements. Let $\mathscr M$ be the set of all $2×2$ matrices with entries in $K$. How many elements of $\mathscr M$ are similar to the following matrix?
$ \begin{pmatrix}
0 & 0 \\
0 & 1
\end{pmatrix}$
My attempt: Answer is given is $56$. We n... | You can continue as below:
$$a-a^2=bc \Longrightarrow a^2-a+bc=0 \Longrightarrow a=\frac{1\pm\sqrt{1-4bc}}{2}\overset{1/2 = 4}{===}4\pm4\sqrt{1-4bc}$$
So $(1-4bc)$ must be square:
$$(1-4bc) \in \{ 0^2,(\pm)1^2,(\pm2)^2,(\pm3)^2 \} = \{ 0,1,4,2 \} \Longrightarrow 4bc \in \{ 1,0,-3,-1 \} \overset{1/4=2}{=\Longrightarrow}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3423194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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If $a, b, c, d \in \mathbb{N}$ and $ab = cd \ne 0$ show that $a^2+b^2+c^2+d^2$ is composite A problem I saw on quora,
slightly modified.
If $a, b, c, d \in \mathbb{N}$
and
$ab = cd \ne 0$
show that
$a^2+b^2+c^2+d^2$
is composite.
My solution uses the theorem that
a number that can be written as the sum of two squares i... | standard is to let $g = \gcd(a,c)$ so that $a = g \alpha$ and $c = g \gamma,$ with
$$ \gcd(\alpha, \gamma) = 1 $$
As $b \alpha = d \gamma$ and $ \gcd(\alpha, \gamma) = 1, $ we find $\gamma|b,$ let
$ b = h \gamma,$ which leads to $d = h \alpha$
$$ a^2 + c^2 + d^2 + b^2 = g^2 \alpha^2 + g^2 \gamma^2 + h^2 \alpha^2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3423653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Is $\frac{x^2}{a^2}+\frac{y^2}{b^2} = \frac{y}{b}$ the equation of an ellipse? Shouldn't it be $\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$?
While solving a question today I came across a locus of the form $$\frac{x^2}{a^2}+\frac{y^2}{b^2} = \frac{y}{b}$$ It was told in my book that it is the equation of an ellipse. How is t... | You can manipulate your equation as so:
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac y b = 0$$
Then we see by completing the square on top of the $y$ terms as below,
$$\begin{align}
\frac{y^2}{b^2} - \frac y b &= \frac{y^2 - by}{b^2} \\
&= \frac{y^2 - by + b^2/4 - b^2/4}{b^2} \\
&= \frac{(y-b/2)^2 - b^2/4}{b^2} \\
&= \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3426761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Let A,B be two $4\times4$ real commuting matrices and $\det(A^2-AB+B^2) = 0$ show that $\det(A+B) + 3\det(A-B) = 6\det(A) + 6\det(B)$ Let $A$, $B$ be two $4\times 4$ matrices with real enteries, such that $AB = BA$ and $\det(A^2-AB+B^2) = 0$. Show that $\det(A+B) + 3\det(A-B) = 6\det(A) + 6\det(B)$.
How can I get start... | We can exploit the fact that each expression is homogeneous on $A$ and $B$.
Assume that $\det(B) \neq 0$. Multiply both sides of the first equation by $\det(B)^{-2}$, and both sides of the second equation by $\det(B)^{-1}$. Using the fact that $A$ and $B$ commute, we have
$$
\det(A^2 - AB + B^2)\det(B)^{-2} = 0 \impli... | {
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closed form for this recurrence I am having trouble figuring out if the following recurrence has a closed form:
$$f(n)=2f(n-1)+(n-1)2^{n-2}$$
I have never really done such problems, so I dont know what is a good strategy. I will appreciate any hints!
Edit: $f$ is defined for $n\geq 1$ and $f(1)=0$.
| I was thinking on using the Z-transform here:
$$ f(n) = 2f(n-1) + (n-1)2^{n-2} $$
using $f(n) = x_n$ and $ f(n-1) = x_{n-1}$ we get:
$$ x_n = 2x_{n-1}+n\frac{2^n}{4}-\frac{2^n}{4} $$
Taking the transform on both sides:
$$ X(z) = 2(z^{-1}X(z)+x_{-1}) + \frac{2z}{4(z-2)^2}-\frac{z}{4(z-2)} $$
$$ X(z) = 2(z^{-1}X(z)+\frac... | {
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"timestamp": "2023-03-29T00:00:00",
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Find $4x \equiv 7 \pmod{15}$ and $3x \equiv 5 \pmod{16}$ (different exercises) This is how I solved each:
$$4x \equiv 7\pmod {15} \\
4x - 7 \equiv 0\pmod{15} \\
4x-7 = 15k \Leftrightarrow 4x-15k= 7 \\
$$
$$15 = 4*3+3 \\
4=3*1+1\\
3=3*1+0$$
$$1 = 4-3*1 \\
3 = 15-4*3 \\
1 = 4 - (15-4*3)*1 \\
1 = 4-15+4*3 \\
1 = 4*4-15*1... | Don't do that with such computations! For each of these congruence equations, you just have to find the modular inverse of the coefficient (which happens to be a modular unit since it is coprime to the modulus).
Example for the first congruence:
The modular inverse of $4\bmod 15$ is none other than itself since $4^2=16... | {
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"source": "stackexchange",
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How do I solve this: I was doing my homework and I stumbled over this particular exercise. I would've known how to solve it, if it had been the same thing under square root in both cases.
$$
\lim_{n\to \infty} \left(\sqrt{4n^2+3n+2}-\sqrt{4n^2+n-1}\right)
$$
| By binomial approximation
$$\sqrt{4n^2+3n+2}=2n\left(1+\frac3{4n}+\frac1{2n^2}\right)^\frac12 \approx 2n+\frac34+\frac1{2n}$$
$$\sqrt{4n^2+n-1}=2n\left(1+\frac1{4n}-\frac1{4n^2}\right)^\frac12 \approx 2n+\frac14-\frac1{4n}$$
therefore
$$\sqrt{4n^2+3n+2}-\sqrt{4n^2+n-1}\approx 2n+\frac34+\frac1{2n}-2n-\frac14+\frac1{4n}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3432776",
"timestamp": "2023-03-29T00:00:00",
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$x,y,z > 0$, prove that $ 3xyz + x^{3}+ y^{3} + z^{3} \ge 2 \left[ (xy)^{3/2} + (yz)^{3/2} + (xz)^{3/2} \right] $ $x,y,z > 0$, prove that
$$ 3xyz + x^{3}+ y^{3} + z^{3} \ge 2 \left[ (xy)^{3/2} + (yz)^{3/2} + (xz)^{3/2} \right] $$
Without using Schur's inequality,
Attempt:
By $C.S$:
$$ (xy)^{3/2} + (yz)^{3/2} + (xz)^{... | First, Schur's inequality provides that
$$
x^3+y^3+z^3+3xyz\ge xy(x+y)+yz(y+z)+zx(z+x).
$$
Then,
$$
xy(x+y)\ge 2xy\sqrt{xy}
$$
and hence
$$
x^3+y^3+z^3+3xyz\ge 2\big(xy\sqrt{xy} +yz\sqrt{yz} +zx\sqrt{zx} \big)
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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How big is the smallest triangle inscribed in a square A square is divided in 7 areas as show on the figure.
The dots show the corners of the square and the middle points on the edges.
How large a fraction is area $D$ and how do I work it out?
I have tried using trigonometry to calculate the area A. If we say each sid... | Alternatively, we can find that the triangle containing $D$ and $E$ has a base of $1$ and a height of $\frac{2}{3}$ (obtained by solving $1-x = \frac{1}{2} + \frac{1}{2}x$), so it has an area of $\frac{1}{2} \times 1 \times \frac{2}{3} = \frac{1}{3}$.
The height of $D$ is $\frac{2}{3} - \frac{1}{2} = \frac{1}{6}$, so t... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find the smallest value of the following expression $\sqrt{(x-9)^2 +4} + \sqrt{x^2+y^2}+ \sqrt{(y-3)^2 +9}$
Find the smallest value of the following expression:
$$\sqrt{(x-9)^2 +4} + \sqrt{x^2+y^2} + \sqrt{(y-3)^2 +9}$$
I tried to derive the expression with respect to $x$ and $y$ and then equal the derivative to ze... | Proceeding as your first idea by derivative, we have that for
$$f(x,y)=\sqrt{(x-9)^2 +4} + \sqrt{x^2+y^2} + \sqrt{(y-3)^2 +9}$$
in order to minimize we can guess that $0<x<9$ and $0<y<3$ and then
$$f_x(x,y)=\frac{x}{\sqrt{x^2 +y^2}}+ \frac{x-9}{\sqrt{(x-9)^2+4}}=0 \implies y^2=\frac{4x^2}{(9-x)^2}\implies y=\frac{2x}{9... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3437202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
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General formula for the power series of $\dfrac{1}{(1+x)^3}$ I wish to find the general formula for the following power series:
$\dfrac{1}{(1+x)^3}=1-3x+6x^2-10x^3+15x^4-21x^5+28x^6...$
The difference between the first and second term is $2$
The difference between the second and third term is $3$
The difference between... | If you know calculus, then
$\frac {1}{(1+x)^3} = \frac {d^2}{dx^2}\frac {1}{2(1+x)} = \frac {d^2}{dx^2} \sum_\limits{n=0}^\infty \frac {(-1)^nx^n}{2} = \sum_\limits{n=0}^\infty \frac {(-1)^n(n+1)(n+2) x^n}{2}$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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$\cot \pi iy$ tends to $-i$ as $y \to \infty$ For a fixed real number $x$, why does $\cot \pi z$ tends to $-i$ as $y \to \infty$? (Here, $z=x+iy$.) Can this be proved simply?
| Note the following:
$\cot(z) = \frac{\cos(z)}{\sin(z)}$
$\cos(x+iy) = \cos(x) \cosh(y) - i \sin(x) \sinh(y)$
$\sin(x+iy) = \sin(x) \cosh(y) + i \cos(x) \sinh(y)$
so
\begin{equation}
\cot(z) = \frac{\cos(x) \cosh(y) - i \sin(x) \sinh(y)}{\sin(x) \cosh(y) + i \cos(x) \sinh(y)}
\end{equation}
and multiply the numerator an... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is the graph of $8x^2+8xy+2y^2+2x+y+5=0$ empty? For the equation
$$8x^2+8xy+2y^2+2x+y+5=0$$
Comparing it with standard equation of conic section we get that
$h^2-ab=0$ and
$$\Delta= \left|\begin{array}{ccc}
a & h & g \\
h & b & f \\
g & f & c \end{array}\right| = 0$$
So as per I know it should represent a pair of... | (as Oscar Lanzi started...)
\begin{align}
2(2x+y)^2 + (2x+y) + 5 &= 0 \\
16(2x+y)^2 + 8(2x+y) &= -40 \\
[4(2x+y)+1]^2 &= -39 \\
\end{align}
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove $S=\{a+b\sqrt[3]{2} +c\sqrt[3]{4}:a,b,c \in \mathbb{Q}\}$ is a ring. Prove $S=\{a+b\sqrt[3]{2} +c\sqrt[3]{4}:a,b,c \in \mathbb{Q}\}$ is a ring.
My professor has gone over how to prove a subring, but I am not sure how to prove a ring.
This is what I have so far, but I am not sure if it is similar to proving a sub... | Alternative proof: $\mathbb{Q}[x]$ is a ring and let $I$ be the ideal generated by $x^3 -2$, then we have that $R = \mathbb{Q}[x]/ I$ is a ring.
We have a homomorphism of rings $\phi: \mathbb{Q}[x] \to \mathbb{R}$ given by $\phi(1) = 1$ and $\phi(x) = \sqrt[3]{2}$, since $\phi(I) \subseteq \{0\}$ this defines a homomo... | {
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If $n\geq 3$, $9^n \equiv a (\mod 100)$ and $9^{n+1} \equiv b (\mod 100)$, then $a+b=90$. I noticed a pattern in the powers of 9 modulo 100.
$9^1 \equiv 9 \pmod{100}$
$9^2 \equiv 81 \pmod{100}$
$9^3 \equiv 29 \pmod{100}$
$9^4 \equiv 61 \pmod{100}$
.
.
.
and conjectured the following:
If $n\geq 3$ is an odd integer wher... | $\ \overbrace{ \color{#c00}{10}\cdot 9^n}^{\textstyle 9^n\!+\!9\cdot9^n\!\!\!\!\!\!\!\!}\bmod \color{#c00}{10}0\, =\, \color{#c00}{10}\,(\!\!\!\overbrace{9^n}^{\large (-1)^{\Large n}}\!\!\!\bmod 10)\, = \,\left\{\begin{align}&10,\,\ n\rm\ even\\ &90,\,\ n\ \rm odd\end{align}\right.$
$\text{using}\,\ \color{#c00}ab\bmo... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What are the values of the parameters $a,b \in \mathbb{R}$ such that $\lim\limits_{x \to \infty}(\sqrt{x^2+x+1}+\sqrt{x^2+2x+2}-ax-b)=0.$ I have to find the values of $a$ and $b$ (with $a,b \in \mathbb{R}$) such that the following is true:
$$\lim\limits_{x \to \infty} (\sqrt{x^2+x+1} + \sqrt{x^2+2x+2}-ax-b)=0$$
This is... | \begin{align*}
&\sqrt{x^{2}+x+1}-x+\sqrt{x^{2}+2x+2}-x\\
&=\dfrac{x+1}{\sqrt{x^{2}+x+1}+x}+\dfrac{2x+2}{\sqrt{x^{2}+2x+2}+x}\\
&\rightarrow \dfrac{1}{2}+\dfrac{2}{2}\\
&=\dfrac{3}{2}.
\end{align*}
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Computing the cycle decomposition of the composition of permutations. Let $\sigma,\tau\in S_5$ be given by
$$
\sigma = \begin{pmatrix}1&3&5\end{pmatrix}\begin{pmatrix}2&4\end{pmatrix},\quad \tau = \begin{pmatrix}1&5 \end{pmatrix}\begin{pmatrix}2&3\end{pmatrix}.
$$
I want to find the cycle decomposition of $\tau\sigma$.... | in Simple Form :
${ \begin{bmatrix}{1} && {2} && {3} && {4} && {5} \\ {5} && {3} && {2} && {4} && {1}\end{bmatrix} }$ ${ \begin{bmatrix}{1} && {2} && {3} && {4} && {5} \\ {3} && {4} && {5} && {2} && {1}\end{bmatrix} }$
= ${ \begin{bmatrix}{1} && {2} && {3} && {4} && {5} \\ {2} && {4} && {1} && {3} && {5}\end{bmatrix}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3448678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Triangle sides $a,b,c$ are in arithmetic progression. Show $\sin^2(A/2)\csc2A$, $\sin^2(B/2)\csc2B$, $\sin^2(C/2)\csc2C$ are in harmonic progression
If sides $a,b,c$ of the triangle ABC are in arithmetic progression, prove that $\sin^2\frac{A}{2}\mathrm{cosec}(2A),\sin^2\frac{B}{2}\mathrm{cosec}(2B),\sin^2\frac{C}{2}\... | You have not used the fact that $a$, $b$, $c$ are in arithmetic progression. WLOG write $a=b-d$, $c=b+d$; then
\begin{align*}
\dfrac{1}{T_1}+\dfrac{1}{T_3}-\dfrac{2}{T_2}&=\dfrac{2}{R}\left(\dfrac{(b^2+c^2-a^2)a}{(a+c-b)(a+b-c)}+\dfrac{(a^2+b^2-c^2)c}{(c+a-b)(c+b-a)}-\dfrac{2(a^2+c^2-b^2)b}{(b+c-a)(a+b-c)}\right)\\
&= ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Find an eigenvector I am trying to find an eigenvector of the following matrix, where I get an eigenvector zero. But, to my knowledge, an eigenvector cannot be zero.
$$ \pmatrix{0&1\\0&i-1}\pmatrix{x\\y} = \pmatrix{0\\0}.$$
| You are looking for an eigenvector $(x,y)$ associated to the eigenvalue $\lambda=0$.
$$
\begin{pmatrix}0&1\\0&i-1\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = 0\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}0\\0\end{pmatrix}
$$
On the other hand,
$$\begin{pmatrix}0&1\\0&i-1\end{pmatrix}\begin{pmatrix}x\\y\end{pma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3451605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Generating function of $a_{n+1} = 4a_{n} - a_{n-1}$ I need to find the generating function of this recurrence when $a_0 = 4$, $a_1$ = 8
$f(x) = \sum_{i=0}^\infty a_ix^i = 4 + 8x + (4a_1 - a_0)x^2 + (4a_2 - a_1)x^3 + \dots$
$f(x) = 4 + 8x + 4a_1x^2 - a_0x^2 + 4a_2x^3 - a_1x^3 + \dots$
$f(x) = 4 + 8x + 4x(a_1x + a_2x^2 +... | To rewrite your work more clearly:
\begin{align}f(x)&=\sum_{n=0}^\infty a_nx^n\\&=4+8x+\sum_{n=0}^\infty a_{n+2}x^{n+2}\\&=4+8x+\sum_{n=0}^\infty(4a_{n+1}-a_n)x^{n+2}\\&=4+8x+4x(f(x)-4)-x^2f(x)\\&=4-8x-(x^2-4x)f(x)\end{align}
$$f(x)=\frac{4-8x}{x^2-4x+1}$$
As noted in the comments, this representation is valid provided... | {
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"timestamp": "2023-03-29T00:00:00",
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$k=-\sqrt{3}(\sqrt{5+2\sqrt{6}}+\sqrt{5-2\sqrt{6}})$
Let $k$ be equal to:
$$k=-\sqrt{3} \left(\sqrt{5+2\sqrt{6}}+\sqrt{5-2\sqrt{6}} \right)$$
I am trying to simplify the expression and express $5+2\sqrt{6}$ and $5-2\sqrt{6}$ as squares. I don't think it's smart to multiply the brackets by $-\sqrt{3}$ at first. Is t... | $\sqrt{5 \pm 2 \sqrt{6}} = \sqrt{3} \pm \sqrt{2}$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Minimise $2x+y$ subject to $x^2+y^2=1$ using KKT I'm doing this exercise in preparing for the final exam in optimization:
$$\begin{align*}
\text{min} &\quad 2x+y \\
\text{s.t} & \quad x^2+y^2=1
\end{align*}$$
Could you please verify if I correctly understanding the KKT's theorem? Thank you so much for your help!
My... | Using that $$(ax+by)^2\le (a^2+b^2)(x^2+y^2)$$ we get
$$(2x+y)^2\le (4+1)(x^2+y^2)=5$$
| {
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"url": "https://math.stackexchange.com/questions/3456783",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find $\lim_{n \to \infty}\frac1{\ln^2n}\left( \frac{\ln 2}{2} + \frac{\ln 3}{3} +\cdots + \frac{\ln n}{n}\right)$ I have to find the following limit:
$$\lim_{n \to \infty} \frac{\frac{\ln 2}{2} + \frac{\ln 3}{3} + \cdots + \frac{\ln n}{n}}{\ln^2n}$$
I tried splitting this limit like so:
$$\lim\limits_{n \to \infty} \... | Define
$$
S_n=\frac{\ln 2}{2} + \frac{\ln 3}{3} + ... + \frac{\ln n}{n}
$$
The function $\ln(x)/x$ is decreasing for $x\geq 3$, so
$$
\frac{\ln 2}{2} + \int_3^n\frac{\ln(x)}{x}\,dx \leq S_n\leq \frac{\ln 2}{2} + \frac{\ln 3}{3} + \int_3^{n}\frac{\ln(x)}{x}\,dx,
$$
$$
\frac{\ln(2)}{2} + \frac{\ln(n)^2}{2}-\frac{\ln(3)^... | {
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Let $f(x)$ be a polynomial such that $f(x)*f(1/x) +3f(x)+3f(1/x)=0$ and $f(3) =24$. What is $f(2) +f(-2)$? Let $f(x)$ be a polynomial such that $f(x)*f(1/x) +3f(x)+3f(1/x)=0$ and $f(3) =24$. What is $f(2) +f(-2)$?
| Let $g(x)=f(x)+3$, then $g(x)g(\frac{1}{x})=9$ and $g(3)=27$.
Let $g(x)=ax^i+... +kx^j$ where $a$ and $k$ are the non-zero coefficients of the lowest and highest powers of $x$ in $g(x)$. Then
$$(ax^i+... +kx^j)(ax^{-i}+... +kx^{-j})=9$$
$$(ax^i+... +kx^j)(k+ ...+ax^{j-i})=9x^j$$
The LHS has a non-zero coefficient of ... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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How can I prove analytically that the golden ratio is less then $\frac{\pi^2}{6}$. Or stated in other terms, prove that
$$\frac{1+\sqrt{5}}{2} < \sum_{n=1}^{\infty}\frac{1}{n^2}$$
| First, note that $\frac{1+\sqrt{5}}{2}$ is a root of the quadratic equation
$$f(x)=x^2-x-1$$
This equation is increasing on $(1/2,\infty)$ which is easily shown as
$$f'(x)=2x-1$$
which has one zero at $x=\frac{1}{2}$ and $f'(1)=1>0$. Now, note that
$$f\left(\frac{81}{50}\right)=\left(\frac{81}{50}\right)^2-\left(\frac{... | {
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$f(x)=x^3-21x+35$ is irreducable over $Q$. Show if $f(\alpha)=0$ then $f(\alpha^2+2\alpha-14)=0$ I am not sure how to do this in a short, nice way.
I have proved it by factoring $f(x)=(x-\alpha)$ and plugging in, but that is very very messy. Is there a slick way?
Thank you!
| We can prove that is $\alpha$ is a root of $x^3-21x+35=0$ then $\alpha^2+2\alpha-14$ is a root of the same equation.
First use completion of squares to render
$\alpha^2+2\alpha-14=(\alpha+1)^2-15$
Now suppose that $\alpha^3-21\alpha+35=0$. Define $A=\alpha+1$:
$(A-1)^3-21(A-1)+35=0$
$A^2-3A^2+3A-1-21A+21+35=0$
$A^3-3A... | {
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"timestamp": "2023-03-29T00:00:00",
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Limit of $\lim_{x \to 0} \frac{(1+x^5)^{10} -1}{(\sqrt{1+x^3}-1)(\sqrt[5]{1+x^2}-1) }$ I tried using symbolab to get the limit of
$$\lim_{x \to 0} \frac{(1+x^5)^{10} -1}{(\sqrt{1+x^3}-1)(\sqrt[5]{1+x^2}-1) }$$
but it couldn't solve it. With WolframAlpha I got $100$ for the limit.
Can someone show me how it's done "per ... | Note that
*
*$\frac{({1+x^k})^{\alpha} - 1}{x^k} \stackrel{t=x^k}{=}\frac{(1+t)^{\alpha} - 1}{t}\stackrel{t\to 0}{\longrightarrow} \left.\left((1+t)^{\alpha}\right)' \right|_{t=0} = \alpha$
Hence,
$$\frac{(1+x^5)^{10} -1}{(\sqrt{1+x^3}-1)(\sqrt[5]{1+x^2}-1) } =\frac{(1+x^5)^{10} -1}{x^{5}}\cdot \frac{1}{\frac{\sqrt{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3469782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
How to find the limit of $a_1=1\:,\:a_{n+1}=\frac{a_n+2}{a_n+1}$ How to find the limit of $a_1=1\:,\:a_{n+1}=\frac{a_n+2}{a_n+1}$?
I think to find the limit, first I need to prove that the sequence is convergent then it is easy to find the limit :
$L = \frac{L+2}{L+1} \to L = \sqrt{2}$ because the sequence $>0$
any... | First, suppose the sequence converges to some $L$. We have
$$a_{n+1}=\frac{a_n+2}{a_n+1}=1+\frac{1}{1+a_n},$$
and we can take the limit of both sides. On the left,
$$\lim_{n\to\infty}(a_{n+1}-L+L)=\lim_{n\to\infty}(a_{n+1}-L)+L=L.$$
Now since $a_n>0$ (or even $a_n>1$) for all $n$, we know that $\lim\limits_{n\to\infty}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3474313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Partial fraction decomposition of $\frac{1}{x^a(x+c)^b}$ We have the partial fraction decomposition
$$\frac{1}{x^a(x+c)^b}=\frac{d_a}{x^a}+\frac{d_{a-1}}{x^{a-1}}+...+\frac{d_{1}}{x}+\frac{e_b}{(x+c)^b}+\frac{e_{b-1}}{(x+c)^{b-1}}+...+\frac{e_{1}}{x+c},$$
where $a,b,c\in\mathbb{N}$.
Is there anything particular we can ... | Multiply both sides by $x^a$ and $(x+ c)^b$ to get $$1= d_a(x+ c)^b+ d_{a-1}x(x+ c)^b+ \cdots+ d_1x^{a-1}(x+ c)^b+ e_bx^a+ e_{b+1}x^a(x+ c)+ \cdots+e_1x^a(x+ c)^{b-1}~.$$
Taking $~x= 0~$ gives immediately $~1= c^b~d_a~$ so $~d_a= 1/c^b~$. Taking $~x= -c~$ immediately gives $~1= e_b~c^a~$ so $~e_b= 1/c^a~$.
That leaves... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3479738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
} |
Why should it be $\sqrt[3]{6+x}=x$? Find all the real solutions to:
$$x^3-\sqrt[3]{6+\sqrt[3]{x+6}}=6$$
Can you confirm the following solution? I don't understand line 3. Why should it be $\sqrt[3]{6+x}=x$?
Thank you.
$$
\begin{align}
x^3-\sqrt[3]{6+\sqrt[3]{x+6}} &= 6 \\
x^3 &= 6+ \sqrt[3]{6+\sqrt[3]{x+6}} \\
x &= ... | I think the step that is is missing is
$$x = \sqrt[3]{6+ \sqrt[3]{6+\sqrt[3]{\color{red}x+6}}}$$
$$=\sqrt[3]{6+ \sqrt[3]{6+\sqrt[3]{6+\color{red}x}}}$$
$$= \sqrt[3]{6+ \sqrt[3]{6+\sqrt[3]{6+\color{red}{\sqrt[3]{6+ \sqrt[3]{6+\sqrt[3]{6+\color{blue}x}}} }}}}, $$
and we can continue to obtain the infinite expression
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3480864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 4,
"answer_id": 0
} |
$\sqrt{m^2 - x^2} \leq x , m > 0$ the range for $x$ is? $\sqrt{m^2 - x^2} \leq x $
$\sqrt{{m}^{2} - {x}^{2}} \geq 0$
$x \geq 0$
$\sqrt{m^2 - x^2} \leq x $
$m^2 - x^2 \leq x^2 $
$m^2 \leq 2x^2$
$\frac{m}{\sqrt 2} \leq x$
$\sqrt{{m}^{2} - {x}^{2}} \geq 0$
${m}^{2} \geq {x}^{2}$
$m > 0, x > 0$
$m \geq x$
$x \leq m$
$\frac... | Let $0<x<m^2$, tthe squaring we get
$$m^2-x^2 \ge x^2 \implies 0\le x \le m/\sqrt{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3481649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find value of $a_1$ such that $a_{101}=5075$ Let $\{a_n\}$ be a sequence of real numbers where $$a_{n+1}=n^2-a_n,\, n=\{1,2,3,...\}$$
Find value of $a_1$ such that $a_{101}=5075$.
I have
$$a_2=1^2-a_1$$
$$a_3=2^2-a_2=2^2-1^2+a_1$$
$$a_4=3^2-2^2+1^2-a_1$$
$$a_5=4^2-3^2+2^2-1^2+a_1$$
$$\vdots$$
$$a_{101}=100^2-99^2+98^... | What if at the point where:
$a_{101} = 100^2 - 99^2 + 98^2 - 97^2 \cdots 2^2 - 1^2 + a_1$
You opted for a clever factorization of squares:
$a_{101} = (100 - 99)(100 + 99) + (98 - 97)(98 + 97) \cdots (2 - 1)(2 + 1) + a_1$
$a_{101} = 199 + 195 + 191 \cdots 3 + a_1$
Therefore those numbers become a simple arithmetic serie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3482467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Find the maximum of a sum Given $x + y + z = 0, x + 1 > 0, y + 1 > 0, z + 4 > 0$, find the maximum of $$Q=\frac{x}{x+1} + \frac{y}{y+1} + \frac{z}{z+4}$$
The answer is $\frac{1}{3}$
I've tried inverting each of the fractions then using AM-GM on them, but it doesn't give me the correct answer
| First set
*
*$a=x+1, b=y+1, c= z+4 \Rightarrow a,b,c >0$ and $a+b+c = 6$
Hence, to maximize is
$$Q = 3-\left(\frac 1a + \frac 1b + \frac 4c \right)$$
So, minimize
$$\left(\frac 1a + \frac 1b + \frac 4c \right) = \frac{a+b+c}6\left(\frac 1a + \frac 1b + \frac 4c\right)$$
$$= \frac 16+\frac 16 + \frac 23 + \frac 16\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3482629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
What's wrong in my calculation of $\int \frac{\sin x}{1 + \sin x} dx$? I have the following function:
$$f: \bigg ( - \dfrac{\pi}{2}, \dfrac{\pi}{2} \bigg ) \rightarrow \mathbb{R} \hspace{2cm} f(x) = \dfrac{\sin x}{1 + \sin x}$$
And I have to find $\displaystyle\int f(x) dx $. This is what I did:
$$\int \dfrac{\sin x}{1... | Beside verifying your answer is equivalent to one of the listed results, you may also integrate in $\tan\frac x2$ since all the choices are in terms of it.
So, use $\sin x =\frac{2\tan\frac x2}{1+\tan^2\frac x2}$ to integrate,
$$\int \dfrac{1}{1 + \sin x}dx = \int \dfrac{1}{1 + \frac{2\tan\frac x2}{1+\tan^2\frac x2}}d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3483283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
number of ways of arranging four pairs of socks in a line so that no adjacent socks form a pair
Suppose there are four pairs of socks, let's label them $AA,BB,CC,DD$. The left and right socks of the pair are indistinguishable. How many ways are there to arrange them in a line such that no adjacent socks are a pair, e.... | We can use the Inclusion-Exclusion Principle.
We have eight positions to fill. We can fill two of them with A's in $\binom{8}{2}$ ways, two of the remaining six positions with B's in $\binom{6}{2}$ ways, two of the remaining four positions with C's in $\binom{4}{2}$ ways, and fill the final two positions with D's in $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3486226",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Suggestions on solving a system of equations How do you suggest solving the following sytem?
$$\begin{cases} \dfrac{12}{x}+\dfrac{12}{y}=1 \\ \dfrac{6}{x-2}+\dfrac{8}{y-6}=\dfrac{2}{3} \end{cases}$$
After simplifying the equations, I got $xy-12y=12x$ and $xy-11y=18x-90$.
Should I continue from here or there's a bette... | Alternatively:
$$\begin{cases} \dfrac{12}{x}+\dfrac{12}{y}=1 \\ \dfrac{6}{x-2}+\dfrac{8}{y-6}=\dfrac{2}{3} \end{cases} \Rightarrow
\begin{cases} \dfrac{12}{x}=1-\dfrac{12}{y} \\ \dfrac{6}{x-2}=\dfrac23-\dfrac{8}{y-6} \end{cases} \Rightarrow
\begin{cases} x=\dfrac{12y}{y-12} \\ x=\dfrac{18(y-6)}{2y-36}+2=\dfrac{22y-18... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3486452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find min and max of $A = \sqrt{x-2} + 2\sqrt{x+1} + 2019 -x$
Find minimum and maximum of $$A = \sqrt{x-2} + 2\sqrt{x+1} + 2019 -x$$
How to solve this problem without using derivatives?
Michael Rozenberg's edit:
Because with a derivative it's not so easy:
$$A'(x)=\frac{1}{2\sqrt{x-2}}+\frac{1}{\sqrt{x+1}}-1=\frac{1... | Let $f(x) = \sqrt{x-2} + 2\sqrt{x+1} + 2019 -x$ then
$$
f'(x) = \frac{1}{2\sqrt{x - 2}} + \frac{1}{\sqrt{x+1}} - 1
$$
Equating this to zero for real $x$, we get $x = 3$. The second derivative is negative hence the global maximum is $f(3) = 2021.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3487710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Prove $2\cos5B+2\cos4B+2\cos3B+2\cos2B+2\cos B+1=\frac{\sin(11B/2)}{{\sin(B/2)}}$ Working through a book, I have stumbled upon a question I don't know how to solve:
Prove
$$2\cos(5B) + 2\cos(4B) + 2\cos(3B) + 2\cos(2B) + 2\cos(B) + 1 = \dfrac{\sin\left(\frac{11B}2\right)}{{\sin\left(\frac B2\right)}}$$
(this is a... | \begin{align}
& 1 + 2\cos B + 2\cos(2B) + 2\cos(3B) + 2\cos(4B) + 2\cos(5B) \\[8pt]
= {} & \cos(-5B) + \cos(-4B) + \cdots + \cos(0B) + \cdots + \cos(4B) + \cos(5B) \\
& \text{(since cosine is an even function)} \\[10pt]
= {} & \operatorname{Re} \big( e^{-5iB} + e^{-4iB} + \cdots + e^{0iB} + \cdots + e^{4iB} + e^{5iB} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3490414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Po-Shen Loh's new way of solving quadratic equation Quadratic equation, $ax^2+bx+c=0$ and its solution is quadratic equation, $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$
Now setting $a=1$ then we have $x^2+bx+c=0$
$$x=\frac{-b\pm \sqrt{b^2-4c}}{2}$$ rewrite as
$$x=-\frac{b}{2}\pm \sqrt{\left(\frac{b}{2}\right)^2-c}$$
In thi... | The teacher's conclusion at the end "guesswork has been replaced by a clever trick" implies that the main result of the proposed new method is the clever trick (change of unknowns $x_1$ and $x_2$) of solving the system of equations (which is the Vieta's theorem):
$$\begin{cases}x_1+x_2=-b\\ x_1x_2=c\end{cases} \stackre... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3490752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 8,
"answer_id": 5
} |
Find $\lim\limits_{n \to \infty} \int_0^1 \sqrt[n]{x^n+(1-x)^n} \,dx$. I have the following limit to find:
$$\lim_{n \to \infty} \int_0^1 \sqrt[n]{x^n+(1-x)^n} \, dx$$
And I have to choose between the following options:
A. $0$
B. $1$
C. $\dfrac{3}{4}$
D. $\dfrac{1}{2}$
E. $\dfrac{1}{4}$
This is what I did:
$$\int_0^1 \... | Let the integrand be $f(x)$. Obviously $f(0)=f(1)=1$ and the function achieves a single minimum $\left(\frac12,\frac{\sqrt[n]2}2\right)$. Hence we have certainly $\frac12<I_n<1$.
If you want to compute the limit anyway,
$$\sqrt[n]{x^n+(1-x)^n}dx=x\sqrt[n]{1+\left(\frac{1-x}x\right)^n}\to x$$ when $1-x<x$, and symmetri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3492509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Evaluate: $\lim\limits_{x \to \infty} (\sqrt{x+2}-\sqrt{x})$
Evaluate: $\displaystyle\lim_{x \to \infty} (\sqrt{x+2}-\sqrt{x})$
I was given this problem. I'm not sure how to tackle it. $\infty-\infty$ is in indeterminate form, so I need to get it into a fraction form in order to solve. I did this by pulling an $x^2$ ... | We have that $\sqrt{x+2} - \sqrt{x} = \dfrac{(\sqrt{x+2}-\sqrt{x})(\sqrt{x+2}+\sqrt{x})}{\sqrt{x+2}+\sqrt{x}} = \dfrac{2}{\sqrt{x-2}+\sqrt{x}}.$ Hence $\lim\limits_{x\to\infty} \sqrt{x+2}-\sqrt{x} = \lim\limits_{x\to\infty}\dfrac{2}{\sqrt{x}(\sqrt{1-\frac{2}{x}}+1)}=0.$
If you were looking for a nonzero answer, you cou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3492588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Why does this pattern occur: $123456789 \times 8 + 9 = 987654321$ I came across the following:
$\begin{align} 1 \times 8 + 1 &= 9 \\ 12 \times 8 + 2 & = 98 \\ 123 \times 8 + 3 & = 987 \\ 1234 \times 8 + 4 & = 9876 \\ 12345 \times 8 + 5 & = 98765 \\ 123456 \times 8 + 6 & = 987654 \\ 1234567 \times 8 + 7 & = 9876543 \\ 1... | $$\left\lfloor {10^n\over (1-x)^2} \right\rfloor \cdot 8+n= 9\cdot \left\lfloor {10^n\over (1-x)^2} \right\rfloor -\left\lfloor {10^n\over(1-x)^2} \right\rfloor +n$$
With $x=1$ is what you've observed ( yes I realize the division by 0, just don't know a better way yet to present what the OP sees). The real question tho... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3492701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 5,
"answer_id": 2
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Value of expression in the roots of the cubic obtained by shifting $x^3-3x^2+x+2$ to eliminate the $x^2$ term
For a given function $f(x)=x^3-3x^2+x+2$, roots are shifted by $\lambda$ to eliminate $x^2$ terms and $\alpha$, $\beta$, $\gamma$ be the roots of new function. Evaluate
$$\begin{align}
\frac{\alpha^3}{(\alpha-... | Hint: $(x^3-2x+1)$=$(x-1)(x^2+x-1)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3494812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Finding kernel and image of endomorphism. I have been trying to solve this, but I can't. I am given the endorphism: $$f(X) = A\cdot X - X\cdot A$$
Where $X$ is any real-valued $2 \times 2$ matrix and $$A = \begin{pmatrix}1 & 2 \\ 2 & -1\end{pmatrix}$$
And I have to find a basis of the kernel and a basis of the image of... | Hint: Observe
\begin{align}
AX-XA =&\
\begin{pmatrix}
1 & 2\\
2 & -1
\end{pmatrix}
\begin{pmatrix}
x & y\\
z & w
\end{pmatrix}
-
\begin{pmatrix}
x & y\\
z & w
\end{pmatrix}
\begin{pmatrix}
1 & 2\\
2 & -1
\end{pmatrix}\\
=&
\begin{pmatrix}
x+2z & y+2w\\
2x-z & 2y-w
\end{pmatrix}
-
\begin{pmatrix}
x+2y & 2x-y\\
z+2w & 2z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3495524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Find the $f^{(3)}(0)$ of $f(x) = \sin^{3}(\ln(1+x))$ using Taylor expansion For the Taylor expansions I used:
$\ln(1+x) = \sum_{k=1}^\infty (-1)^{k+1} \frac{x^{k}}{k}$
$\sin(x) = \sum_{k=0}^\infty (-1)^{k+1} \frac{x^{2k+1}}{(2k+1)!}$
So $\sin(x) = x - \frac{x^{3}}{3!} + O(x^{5})$
and $\ln(1+x) = x - \frac{x^2}{2} + \fr... | This is correct, though take note that you should be more careful with your remainders on each step to ensure you didn't miss anything that might contribute to the $x^3$ coefficient:
\begin{align}\sin(\ln(1+x))&=\sin\bigg(x-\frac{x^2}2+\frac{x^3}3+\mathcal O(x^4)\bigg)\tag{$\star$}\\&=\left[\color{#2255FF}{x-\frac{x^2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3495761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How to factorize $X^5-1$ into irreductible factors? I have to factorize the polynomial $P(X)=X^5-1$ into irreducible factors in $\mathbb{C}$ and in $\mathbb{R}$, this factorisation happens with the $5$th roots of the unity.
In $\mathbb{C}[X]$ we have $P(X)=\prod_{k=0}^4 (X-e^\tfrac{2ki\pi}{5})$.
In $\mathbb{R}[X]$ the... | In $\mathbb C[x]$ we can write it as $$p(x)=(x-1)(x-\color{blue}{e^{2\pi i/5}})(x-\color{blue}{e^{-2\pi i/5}})(x-\color{red}{e^{4\pi i/5}})(x-\color{red}{e^{-4\pi i/5}})$$ where the blue and red are conjugate root pairs. Multiplying these conjugate root pairs together gives (i.e. multiply the second and third terms tog... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3496594",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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How to evaluate $\int_{0}^{\pi}\frac{2x^3-3\pi x^2}{(1+\sin{x})^2}\,\textrm{d}x\,$? Problem:
Let $f$ be a bounded continuous function on the interval [0,1].
(a) show that $\int_{0}^{\pi} xf\,(\sin{x})\,\textrm{d}x = \frac{\pi}{2}\int_{0}^{\pi}f\,(\sin{x})\,\textrm{d}x\,$.
(b) Hence evaluate $\int_{0}^{\pi} \dfrac{x}{1+... | Here is to derive the last integral below using the result in Part (b).
$$J=\int_0^\pi \frac{dx}{(1 + \sin x)^2}
=\int_0^\pi d\left(-\frac{\cos x}{1 + \sin x}\right)\frac{1}{1 + \sin x}$$
Integrate by parts,
$$J=2-\int_0^\pi \frac{\cos^2 x}{(1 + \sin x)^3}dx
= 2- \int_0^\pi \frac{1 - \sin x}{(1 + \sin x)^2}
=2+\int_0^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3498840",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
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sum of coefficient of all even power of $x$
The sum of all Coefficient of even power of $x$ in
$(1-x+x^2-x^3+\cdots +x^{2n})(1+x+x^2+x^3+\cdots +x^{2n}),n\in \mathbb{N}$
what i try
for $n=1,$ we have
$(1-x+x^2)(1+x+x^2)=(1+1)-(1)+(1+1)=3$
for $n=2,$ we have
$(1-x+x^2-x^3+x^4)(1+x+x^2+x^3+x^4)$
$=(1+1+1)-(1+1)+(1+1+1)... | Let's call sum of odd coefficients $C_O$ and sum of even coefficients $C_E$. You can write your polynomial as $$P(x)=O(x)+E(x)$$, where $O(x)$ is the sum of odd powered terms and $E(x)$ is the sum of even powered terms. Note that if you plug in $x=1$, you get $$C_O=O(x=1)$$ and $$C_E=E(x=1)$$ Similarly, if you plug in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3500198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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$\begin{cases} x^3-y^3=19(x-y) \\ x^3+y^3=7(x+y) \end{cases}$
I should solve the following system: $$\begin{cases} x^3-y^3=19(x-y) \\
x^3+y^3=7(x+y) \end{cases}$$
by reducing the system to a system of second degree.
We can factor:
$$\begin{cases} (x-y)(x^2+xy+y^2)=19(x-y) \\
(x+y)(x^2-xy+y^2)=7(x+y) \end{cases}... | To avoid all possible problems, add the two equations to get
$$x^3-13 x+6 y=0 \implies y=\frac{1}{6} \left(13 x-x^3\right)$$ Plug in the first equation to end with
$$x \left(x^8-39 x^6+507 x^4-2665 x^2+4788\right)=0$$ So $x=0$ if a root.
For the remaining, let $z=x^2$ to make
$$z^4-39z^3+507z^2-2665z+4788=0$$ By inspec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3500828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Solving differential equation by separating variables Solve the following differential equation:
$$\frac yx \frac {dy}{dx}=\sqrt {1+x^2+y^2+x^2y^2}$$
I have tried like this:
$$\frac yx \frac{dy}{dx}=\sqrt {1+x^2+y^2(1+x^2)}\\
\implies \frac yx \frac {dy}{dx}=\sqrt {(1+x^2)(1+y^2)}\\
\implies \int \frac {ydy}{\sqrt {1+y... | It should be like this:
$$\int x\sqrt{1+x^2} dx=\int \dfrac{y}{\sqrt{1+y^2}}dy$$
Also,
$$\int \dfrac{y}{\sqrt{1+y^2}}dy=\sqrt{1+y^2}+C$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3508437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Find eigenvalues of a linear map knowing its eigenvectors Let $\;f: R^{3} \rightarrow R^{3}\;$ be an endomorphism whose eigenvectors are $(0,1,-2),(1,0,4),(1,0,-2)$. Knowing that $f(0,1,0) = (2,1,2)$ find the eigenvalues of $f$.
What is the methodology to solve this problem?
Thank you.
| A more pedestrian approach is to use the matrix for $f$. The condition that $f(0,1,0)=(2,1,2)$ means that the middle column of the matrix must be $(2,1,2)$.
Also $f(1,0,4)=(.,0,.)$ and $f(1,0,-2)=(.,0,.)$ so the middle row of the matrix must be 0,1,0. So we can write the matrix as
$$\begin{pmatrix}a&2&b\\
0&1&0\\
c&2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3508932",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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A circle is drawn with its centre on the line $x + y = 2$ to touch the line $4x – 3y + 4 = 0$ and pass through the point $(0, 1)$. Find its equation. A circle is drawn with its centre on the line $x + y = 2$ to touch the line $4x – 3y + 4 = 0$ and pass through the point $(0, 1)$. Find its equation.
My attempt is as fol... | Let $(a, 2 - a)$ be a center of circle A.
Line $L_0 : 4x - 3y + 4 = 0$ is tangent of A, so point of tangency P has minimum distance with A.
For all X = $(x, y) \in L_0$,
$$
\begin{align*}
d (X, A)^2 = \left( 4^2 + 3^2 \right) \left( \left( x - a \right)^2 + \left(y - 2 + a\right)^2 \right) & \geq \left( 4 \left(x - a\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3510191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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How to solve $\sin 2x \sin x+(\cos x)^2 = \sin 5x \sin 4x+(\cos 4x)^2$? How to solve $\sin 2x \sin x+(\cos x)^2 = \sin 5x \sin 4x+(\cos 4x)^2$?
\begin{align*}
2\cos x(\sin x)^2+(\cos x)^2 & = \frac{1}{2}(\cos x- \cos 9x) +(\cos 4x)^2\\
4\cos x(1-(\cos x)^2)+2(\cos x)^2 & = \cos x + \cos 9x +(2(\cos 4x)^2-1)+1\\
\cos x(... | You should be able to transform the equation into polynomial of $\cos(x)$.
For example, $\sin(2x)\sin(x)=2\sin^2(x)\cos(x)=2(1-\cos^2(x))\cos(x)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3511897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
solve for $y$ in $e^{xy}+x^2+y-1.2$
I cant figure out how to make $y$ the subject in this equation:
$$e^{xy}+x^2+y-1.2=0$$
I did these steps,
$$\begin{align}&y=1.2-x^2-e^{xy}\\
&\ln(y)=\ln(1.2)-\ln(x^2)-\ln(e^{xy})\\
&\ln(y)=\ln(1.2)-\ln(x^2)-xy\\
&\ln(y)+xy=\ln\frac{1.2}{x^2}\end{align}$$
ahead of this I am lost ... | Let $a=1.2$, rearrange a little and we have
\begin{eqnarray*}
e^{-xy} (a-x^2-y)=1.
\end{eqnarray*}
Multiply by $xe^{ax-x^3}$
\begin{eqnarray*}
(\color{red}{ax-x^3-xy})e^{\color{red}{ax-x^3-xy}}=xe^{ax-x^3}.
\end{eqnarray*}
Now recall the Lambert $W$ function is defined by $\color{red}we^{\color{red}w}=z$ gives $w=W(z)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3511979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find the volume cut off from the sphere $x^2+y^2+z^2=a^2$ by the cylinder $x^2+y^2=ax$
Find the volume cut off from the sphere $x^2+y^2+z^2=a^2$ by the cylinder $x^2+y^2=ax$
Attempt: The projection of the Cylinder ( denoted $D$) on the $xy$ plane is a circle which has the equation: $x^2+y^2=ax ~~~\equiv~~~(x-a/2)^2+y... | $V=2\int_{-\pi/2}^{\pi/2} \int_{0}^{a \cos \theta} \sqrt {a^2-r^2}~~ r~dr~ d\theta\\
V=2\int_{-\pi/2}^{\pi/2}-\frac 13 (a^2-r^2)^\frac 32|_0^{a\cos\theta}~~ d\theta$
When we evaluate the limit at $a\cos \theta$....
$(a^2-a^2\cos^2\theta)^\frac 32\\
(a^2\sin^2\theta)\frac 32\\
|a^3\sin^3\theta|$
That is we only consider... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3513290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
On the Diophantine equation $m^2 - p^k = 4z$, where $z \in \mathbb{N}$ and $p$ is a prime satisfying $p \equiv k \equiv 1 \pmod 4$ The title says it all:
Do there always exist $m, p, k \in \mathbb{N}$ such that $m^2 - p^k = 4z$, $z \in \mathbb{N}$ and $p$ is prime satisfying $p \equiv k \equiv 1 \pmod 4$?
MY ATTEMPT ... | Consider the case $z = 4$.
We obtain
$$m^2 - p^k = 16$$
$$m^2 - 16 = p^k$$
$$(m + 4)(m - 4) = p^k$$
$$p^{k-x} = m + 4$$
$$p^x = m - 4$$
$$p^{k-x} - p^x = 8$$
$$p^x (p^{k-2x} - 1) = 8$$
Since $p$ is prime and $p \equiv 1 \pmod 4$, then we have $\gcd(p^x, 8) = 1$, which implies that
$$p^{k-2x} - 1 = 8$$
$$p^{k-2x} = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3515496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Finding the third roots of unity by equating $(a + bi)^3$ to $1$ I know what the third roots of unity are but I want to solve this exercise:
Complex numbers can be written as $a + bi$. Simplify $(a + bi)^3$ and
equate the real part to $1$ and the imaginary part to $0$ to find the
three roots of unity.
So I expand... | Factoring a difference of cubes, we have
\begin{align}
(a+bi)^3&=1\\
(a+bi)^3-1&=0\\
(a+bi-1)((a+bi)^2+(a+bi)+1)&=0\\
(a-1+bi)(a^2+2abi-b^2+a+bi+1)&=0.
\end{align}
So, we have either $$\operatorname{Re}(a-1+bi)=1\ \wedge\ \operatorname{Im}(a-1+bi)=0 \implies (a,b)=(1,0)$$
or
$$\operatorname{Re}(a^2+2abi-b^2+a+bi+1)=1\ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3517726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
Cyclic Inequality $(x^8+1)(x^4+1)(y^8+1)(y^4+1)(z^8+1)(z^4+1)≥((x^2+1)(y^2+1)(z^2+1))^2$ I have been challenged to prove the following:
If $xy+yz+zx=3$ for positive $x,y,z$, then $(x^8+1)(x^4+1)(y^8+1)(y^4+1)(z^8+1)(z^4+1)≥((x^2+1)(y^2+1)(z^2+1))^2$.
Initial attempts at AM-GM are not very helpful for me, I don't think.... | We begin with the following observation $$(x^8+1)(x^4+1)=((x^4)^2+1^2)((x^2)^2+1^2)\geq(x^4\cdot x^2+1\cdot1)^2=(x^6+1)^2.$$
By the Cauchy-Schwarz Inequality $$\prod_{cyc}(x^8+1)(x^4+1)\geq\prod_{cyc}(x^6+1)^2.$$
Thus, it's enough to prove that
$$\prod_{cyc}(x^4-x^2+1)\geq1$$ and since $$x^4-x^2+1\geq\left(\frac{x^2+1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3521631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
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.