Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Number of integer solutions of the equation $x + y + z = 30$ if $x \geq 2$, $y \geq 0$, $z \geq -3$ I am trying to solve a problem using permutation/combination but cannot figure out how to proceed.
Suppose the sum of three variables $x, y, z$ is $30$. If $x\ge2, y\ge0, z\ge-3$, how many integer solutions exist?
I un... | We wish to find the number of solutions of the equation
$$x + y + z = 30 \tag{1}$$
in the integers subject to the restrictions that $x \geq 2$, $y \geq 0$, and $z \geq -3$.
Let $x' = x - 2$, $y' = y$, and $z' = z + 3$. Then $x', y', z'$ are nonnegative integers. Substituting $x' + 2$ for $x$, $y'$ for $y$, and $z' ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2909070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Finding joint CDF from joint PDF.
Suppose $X$ and $Y$ are random variables with joint density
$$
f(x,y) = \left\{
\begin{array}{ll}
1/\pi & \quad\text{if}\ x^{2} + y^{2} \leq 1 \\
0 & \quad \mathrm {otherwise}
\end{array}
\right.$$
Find the joint CDF of $X$ and $Y$.
Ho... | I think the joint CDF $F(x,y)= \mathbb P(X \le x,Y \le Y)$ may be fairly involved, and looking at this diagram suggests the following to me:
*
*If $x \le -1$, or if $y \le -1$, or if $x \le 0$ and $y \le 0$ and $x^2+y^2 \ge 1$:
$$F(x,y)=0$$
*
*If $x \ge 1$ and $y \ge 1$:
$$F(x,y)=1$$
*
*If $-1 \le x \l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2909296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Can we prove this inequality $1+\sqrt{2}+\cdots+\sqrt{n}<\frac{4n+3}{6}\sqrt{n}$ using integrals by parts? Can we prove this inequality $1+\sqrt{2}+\cdots+\sqrt{n}<\dfrac{4n+3}{6}\sqrt{n}$ using integrals by parts?
I need to prove this inequality, and I tried this way:
$$\begin{split}
\int_0^n\sqrt{x} \,dx &=\sum\limit... | First, see that we have
$$
\int_0^n \sqrt{x} \;dx = \frac{4n}{6}\sqrt{n}
$$
which gives us a "part" of the demanded result. Moreover, it suggests that we should try to prove the following:
$$
\sum_{k=1}^n \int_0^1 \frac{t}{2\sqrt{t+k-1}} \;dt < \frac{3}{6}\sqrt{n} = \frac{1}{2}\sqrt{n}
$$
Evaluating the integral we ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2912228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Finding a closed form for $\sum_{n = 1}^{\infty} \frac{1}{(1 + x^2)^{n}}$ So I got the sum
$$\sum_{n = 1}^{\infty} \frac{1}{(1 + x^2)^{n}}$$
and I want a closed form.
Can I just do
$$\frac{1}{1 - r} = \frac{1}{1 - \frac{1}{1 + x^2}} = \frac{1}{x^2} + 1 $$
for a closed form?
| The formula that you are using for the sum of a geometric series, namely that
$$
\sum_{n=0}^\infty ar^n = \frac{a}{1-r},
$$
is valid only when $|r|<1$. In your case, the common ratio $r$ is
$$
\frac{1}{1+x^2},
$$
which is less than $1$ for all $x \neq 0$. So, for $x \neq 0$, you can express your series as
$$
\frac{\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2913996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Determinant of Symmetric Matrix $\mathbf{G}=a\mathbf{I}+b\boldsymbol{ee}^T$ To give the close-form of $\det(\mathbf{G})$, where $\mathbf{G}$ is
\begin{align}
\mathbf{G}=a\mathbf{I}+b\boldsymbol{ee}^T
\end{align}
in which $a$ and $b$ are constant, and $\boldsymbol{e}$ is a column vector with all elements being $1$. In ... | A More Basic Approach
Let us consider for illustration, $G_4$:
$$
G=aI_{4\times4}+bee^T \\
\text{where } e = \begin{bmatrix} 1\\ 1\\ 1\\ 1 \end{bmatrix}
\\
\text{Hence, } G =
\begin{bmatrix}
a & 0 & 0 & 0 \\
0 & a & 0 & 0 \\
0 & 0 & a & 0 \\
0 & 0 & 0 & a \\
\end{bmatrix} +
\begin{bmatrix}
b & b ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2916569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
How to evaluate $1+\frac{2^2}{3!}+\frac{3^2}{5!}+\frac{4^2}{7!}+\cdots$? I learnt that $\displaystyle \sum_{n=0}^{\infty} \frac{n+1}{(2n+1)!} = \frac{e}{2}$.
I am wondering what the closed form for $\displaystyle \sum_{n=0}^{\infty} \frac{(n+1)^2}{(2n+1)!}$ is.
I tried using the fact that $ 1+3+5+\cdots+(2n-1) = n^2$,... | Note that
$$\sinh x = \frac{e^x - e^{-x}}{2} = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots$$
Then
$$x \sinh x = x^2 + \frac{x^4}{3!} + \frac{x^6}{5!} + \cdots$$
Now we replace $x$ with $\sqrt{x}$ getting
$$\sqrt{x} \sinh \sqrt{x} = x + \frac{x^2}{3!} + \frac{x^3}{5!} + \cdots$$
We take the derivative
$$(\sqrt{x} \sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2917988",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
} |
Limit of quotient I recently learned that when you are solving for the limit of a quotient, you have to divide everything by the highest number in the denominator, like this
$$ \lim_{x \to \infty} \frac{\sqrt{4 x^2 - 4}}{x+5} = \lim_{x \to \infty} \frac{\sqrt{4 - \frac{4}{x^2}}}{1 + \frac{5}{x}} = 2.$$
But I don't quit... | $\dfrac{\sqrt{4 \cdot 5^2}}{5}
= \dfrac{\sqrt{4 \cdot 5^2}}{\sqrt{5^2}}
= \sqrt{\dfrac{4 \cdot 5^2}{5^2}}
= 2$
$\dfrac{\sqrt{4 \cdot (-5)^2}}{-5}
= \dfrac{\sqrt{4 \cdot (-5)^2}}{-\sqrt{(-5)^2}}
= -\sqrt{\dfrac{4 \cdot (-5)^2}{(-5)^2}}
= -2$
If $x < 0$, then $x = -\sqrt{x^2}$.
$\dfrac{\sqrt{4 \cdot x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2918223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Getting a closed form from $\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \sum_{k=1}^{j} 1$ I need to get a closed form from
$$
\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \sum_{k=1}^{j} 1
$$
Starting from the most outer summation, I got
$$
\sum_{k=1}^{j} 1 = j
$$
But now I don't know how to proceed with:
$$
\sum_{i=1}^{n-1} \sum_{j=i+1}^{n... | Thanks @Winther for pointing out the previous mistake
\begin{equation}
\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \sum_{k=1}^{j} 1
\end{equation}
We know that
\begin{equation}
\sum_{k=1}^{j} 1 = j
\end{equation}
So
\begin{equation}
\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \sum_{k=1}^{j} 1 =
\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} j
\en... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2918464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
If $x>0$ real number and $n>1$ integer, then $(1+x)^n>\frac{1}{2}n(n-1)x^2$ If $x>0$ real number and $n>1$ integer, then $(1+x)^n>\frac{1}{2}n(n-1)x^2$.
Is there a way to prove it without using the Binomial Theorem? Is it possible to use the Bernoulli's Inequality to prove it? If yes, please show me.
I tried to prove... | Let, $f(x)=(1+x)^n-\frac{1}{2} n(n-1) x^2.$
So, $f'(x)=n(1+x)^{n-1}-n(n-1)x>0 \forall n>1$
Since,for $n>1$ and $x\ge 0,$ by A.M.-G.M. inequality we have,
$$\frac{1.(1+x)^{n-1}+(n-2).1}{n-1}>(1+x)$$
$$\implies (1+x)^{n-1}>(n-1)x$$
Therefore, $f$ is strictly increasing.
Hence we have,
$$f(x)>f(0)\implies (1+x)^n>\frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2919501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Calculating the sum of the infinite series $\frac{1}{5} + \frac{1}{3}\frac{1}{5^3} + \frac{1}{5} \frac{1}{5^5} +......$ How do I calculate the sum of the infinite series?
$$\frac{1}{5} + \frac{1}{3}.\frac{1}{5^3} + \frac{1}{5}. \frac{1}{5^5} +......$$
My attempt :
I know that $$\log (\frac{1+x}{1-x}) = 2 \, \left(x ... | From your own formula,
$$\frac15+\frac13\frac1{5^3}+\frac15\frac1{5^5}+\cdots=\frac12\log\frac{1+\dfrac15}{1-\dfrac15}=\frac12\log\frac32.$$
Check:
The series is fast converging. With the first five terms, $0.202732552127\cdots$, vs. the exact value $0.2027325540541\cdots$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2925053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
No of matrices of rank 3 I was tackling this problem
How many 4 x 3 matrices can be formed of rank 3 where entries are coming from a field of 3 elements.
When I solved it as finding total no of linearly independent sets with cardinality 4 in F^4(F). Assuming that each column of the matrix is coming from F^4(F).
I am ... | You're only counting matrices where the upper $3\times3$ part has full rank. You need to add three more cases: If dependence first occurs in the third row, we have $(3^3-1)(3^3-3)3^2(3^3-3^2)$ options; if it first occurs in the second row, we have $(3^3-1)3(3^3-3)(3^3-3^2)$ options, and if it first occurs in the first ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2927242",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
show that $ \ a_n \leq x \leq a_n+\frac{1}{b^n} \ $ for any base $b \neq 10 $ holds for any real number in $[0,1]$. Every infinite sequence $ \ 0. a_1a_2a_3 \cdots \cdots $ represents a real number $ \ x \in [0,1] $ and $ \ \sum\limits_{i=1}^{n} \frac{a_i}{10^i} \leq x \leq \sum\limits_{i=1}^{n} \frac{a_i}{10^i} +\fr... | Take the difference
$$0\leq x - \sum\limits_{i=1}^{n} \frac{a_i}{b^i} = \sum\limits_{i=n+1}^{\infty} \frac{a_i}{b^i} \leq ...$$
and because $0\leq a_i \leq b-1$ and assuming an integer base $b>1$, we have
$$ ... \leq \sum\limits_{i=n+1}^{\infty} \frac{b-1}{b^i}=
\frac{b-1}{b^{n+1}} \left(\color{red}{\sum\limits_{i=0}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2928355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solving $\sqrt{8-x^2}-\sqrt{25-x^2}\geq x$
I would like to find the solution of
$$\sqrt{8-x^2}-\sqrt{25-x^2}\geq x.$$
My try:
First I used the hint of this answer.
$$ \frac{8-x^2-25+x^2}{\sqrt{8-x^2}+\sqrt{25-x^2}}\geq x \leftrightarrow \frac{-17}{\sqrt{8-x^2}+\sqrt{25-x^2}}\geq x.$$
Then the solution can be found by... | Dinesh's answer is alright, but there is another approach. Your question can be rephrased as finding the range of $x$ such that $f(x)\geq 0$, where
$$f(x) = \sqrt{8-x^2}-\sqrt{25-x^2}-x.$$
We find the derivative of $f$ then set it to be $0$ to find its extreme values:
$$ f'(x) = -\frac{x}{\sqrt{8-x^2}}+\frac{x}{\sqrt{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2929463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Positive Definite Matrix (Block Matrix) Let $B$ be an $m\times n$ matrix. Is $$ A=\begin{pmatrix} I & B \\ B^T & I+B^TB \end{pmatrix} $$ positive definite?
Attempt:
Let $\mathbf{z}=\begin{pmatrix} \mathbf{x} \\ \mathbf{y} \end{pmatrix}$. To show that $A$ is positive definite, $\mathbf{z}^TA\mathbf{z}>0$. Expanding $\ma... | Sylvester's Law of Inertia.
We have your symmetric $A.$ We are going to construct $P^T A P = D$ where $\det P \neq 0$ and $D$ is diagonal. Then the count of positive eigenvalues for $A$ is the same as the count of positive eigenvalues of $D$
$$
\left(
\begin{array}{cc}
I&0 \\
-B^T& I \\
\end{array}
\right)
\left(
\beg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2929861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
For positve $a$, $b$, $c$, $d$ with $a+b+c+d\leq 1$, prove that $\sqrt[4]{(1-a^4)(1-b^4)(1-c^4)(1-d^4)}\geq255\cdot a b c d .$
Let $a,b,c,d\in\mathbb R_+$ such that $a+b+c+d\leqslant1$. Prove that$$
\sqrt[4]{\smash[b]{(1-a^4)(1-b^4)(1-c^4)(1-d^4)}}\geqslant255·abcd.
$$
My observations:
I can see that all of $a,b,c,d$... | By AM-GM we obtain:
$$\sqrt[4]{\prod\limits_{cyc}(1-a^4)}\geq\sqrt[4]{\prod\limits_{cyc}((a+b+c+d)^4-a^4)}=$$
$$=\sqrt[4]{\prod_{cyc}\left((b+c+d)\left((a+b+c+d)^3+(a+b+c+d)^2a+(a+b+c+d)a^2+a^3\right)\right)}\geq$$
$$\geq\sqrt[4]{\prod_{cyc}\left(3\sqrt[3]{bcd}\left(64\sqrt[4]{a^3b^3c^3d^3}+16a\sqrt{abcd}+4\sqrt[4]{abc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2930035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Miscalculating the determinant I am learning linear algebra and am getting stuck when trying to calculate the determinant using elementary row operations. Consider the matrix A.
\begin{vmatrix}
0 & 1 & 2 & 3 \\
1 & 1 & 1 & 1 \\
-2 & -2 & 3 & 3 \\
1 & 2 & -2 & -3 \\
\end{vmatrix}
According to the solution in my textbook... | Actually, the first thing that you did was to exchange rows $1$ and $4$, not $1$ and $3$. After that, when you subtracted row $1$ from row $2$, you should have got a $4$ at the end of the row. And when you add twice the first row to the third one, the last entry should become $-3$, not $9$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2934676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Solution for $\frac{3}{x-9} \gt \frac{2}{x+2}$ What I did:
$\frac{3}{x-9} \gt \frac{2}{x+2}$
$3(x+2) \gt 2(x-9)$
$3x+6 \gt 2x-18$
$x \gt -24$
When typing this in in symbolab, it showed me that the solution is $-24 \lt x \lt -2$ or $x \gt 9$
What did i do wrong ?
How come i didnt get the correct solution ?
| Consider the function $f(x) = \dfrac{3}{x-9} - \dfrac{2}{x+2}$.
The function $f(x)$ has a zero at $x=-24$ and poles at $x=-2$ and at $x=9$. The two poles and the one zero divide the $x-$axis into four intervals
$$(-\infty, -24), \ (-24, -2), \ (-2, 9), (9, \infty) $$
In each of those intervals, $f(x)$ is either entirel... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2934788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Prove by induction that, for all positive integers n, the following inequality holds: (1-2-1)(1-2-2)...(1-2-n) ≥ 1/4 + 2-(n+1)
My solution so far:
After verifying the base case for n = 1, I began the inductive step:
(1-2-1)(1-2-2)...(1-2-(k+1)) ≥ 1/4 + 2-(k+2)
Subtracting 1/4 + 2-(k+1) on the RHS as it's an inequality:... | Assume the inequality holds for some $n = k$, $k \ge 1$,
$$\left(1-2^{-1}\right)\left(1-2^{-2}\right)\cdots\left(1-2^{-k}\right) \ge \frac14 + 2^{-(k+1)}$$
For $n = k+1$,
$$\begin{align*}
LHS &= \left(1-2^{-1}\right)\left(1-2^{-2}\right)\cdots\left(1-2^{-k}\right)\left(1-2^{-(k+1)}\right)\\
&\ge \left(\frac14 + 2^{-(k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2937110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Showing that $\sum_{x \in \mathbb{F}_p} \left(\frac{x^2-1}{p}\right) = -1$, where $\left(\frac{x}{p}\right)$ is the Legendre symbol The question is,
Show that $$\sum_{x \in \mathbb{F}_p} \left(\frac{x^2-1}{p}\right) = -1$$ where the operation $\left(\frac{x}{p}\right) = \pm 1$ if $x$ is a quadratic residue/non-residu... | This is my attempt at what I believe is a full answer following the hints given in the comments.
We first show that there are exactly $p-1$ solutions to the equation $$y^2 = x^2 - 1$$ for pairs $(x,y)\in \mathbb{F}_p^2$. If $y^2 = x^2 -1$ then $(x-y)(x+y) = 1$. Letting $$u = x-y$$ $$v = x+y$$ we see that for any choice... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2939750",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Given two points that are joined by a line that is a tangent to a curve, find the missing constant in the equation for the curve How do I solve a question like this?
I have been given two points that make up a line tangent to $y=\frac{a}{(x+2)^2}$. I need to determine $a$.
First, I calculated the equation of the tangen... | The tangent point and slope there are the same for both equations.
Therefore $mx + c = \frac{a}{(x+2)^2}$ and $m = -\frac{2a}{(x+2)^3}$ for a
certain $x$ value.
To find the x value, set $(mx + c)(x+2)^2 = a$
And $m(x+2)^3 = -2a$
So $-\frac{m}{2}(x+2)^3 = (mx + c)(x+2)^2$
$-\frac{m}{2}(x+2) = mx + c$
$-\frac{mx}{2} - m ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2940026",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Evaluating $\mathcal{P}\int_0^\infty \frac{dx}{(1 + a^2x^2)(x^2 - b^2)}$ I am trying to evaluate this integral
$$\mathcal{P}\int_0^\infty \frac{dx}{(1 + a^2x^2)(x^2 - b^2)}$$
with $\mathcal{P}$ the principal value and $a,b>0$.
I already know the answer to be $$ - \frac{a\pi}{2(1 + a^2b^2)}$$
after fiddling with Mathem... | Use that $$\frac{1}{(1+a^2x^2)(x^2-b^2)}=1/2\,{\frac {1}{ \left( 1+{a}^{2}{b}^{2} \right) b \left( -b+x
\right) }}-{\frac {{a}^{2}}{ \left( 1+{a}^{2}{b}^{2} \right) \left(
1+{a}^{2}{x}^{2} \right) }}-1/2\,{\frac {1}{ \left( 1+{a}^{2}{b}^{2}
\right) b \left( b+x \right) }}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2941019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Prove by the limit definition that $\lim_{x\rightarrow 2}\frac{x^2-1}{x-3}=-3$ I was reviewing $\mathbb{R}-$analisys with a friend and I'm thinking about one of the questions...
Prove by the $\epsilon-\delta$ limit definition that $\lim_{x\rightarrow 2}\frac{x^2-1}{x-3}=-3$.
My answer was very long, someone could do ... | Hint: Recall that $0<a<b$ implies that $\frac{1}{a}>\frac{1}{b}>0$. So,
$$0<\vert x-2 \vert<\delta$$
$$\Longrightarrow \ 1<\vert x-2 \vert+1<\delta+1 $$
$$\Longrightarrow \ 1>\frac{1}{\vert x-2\vert+1}>\frac{1}{\delta+1} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2942204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Is there any other way to solve this difficult integral $ \int \frac{1}{ (a^2 \cos ^2 x + b ^ 2 \sin ^2x) ^2} \ dx $ $$
\int \frac{1}{ (a^2 \cos ^2 x + b ^ 2 \sin ^2x) ^2} \ dx
$$
So this is the question .
The solution given in book is to divide numerator and denominator by $\cos ^4x$ and then substitute $\tan x = t... | Note that
$$I(p,q)=\int \frac{dx}{ a^2\cos ^2 x + b^2 \sin ^2x}
=\int \frac{d(\tan x)}{ a^2+b^2\tan ^2 x }
=\frac1{ab}\tan^{-1}\bigg({\frac ba}\tan x \bigg)
$$
and evaluate
\begin{align}
&\int \frac{1}{ (a^2\cos ^2 x + b^2 \sin ^2x)^2}dx
= -\frac12\left(\frac{I’_a}a+\frac{I’_b}b\right)\\
=&\ \frac1{2ab}\left(\frac1{a^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2943314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Tournament probability question If we have 8 teams in the quarter final
if 3 teams are from the same country
what is the probability that 2 of the 3 will be competing against each other in the quarter final?
From my understanding, if we only had 2 team, probability is basically 1/7 because we are drawing 1 from the rem... | It's the number of $4$ match sets containing a pairing of $2$ of the $3$ teams from the same country divided by the total number of $4$ match sets.
We can number teams $1$ to $8$ with $1$ to $3$ being from the same country. For the number of $4$ match sets containing a pairing of $2$ of the $3$ teams from the same coun... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2943541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
product of two Jacobi matrices Let
$$
A = \begin{bmatrix}
a_1 & b_1 \\
b_1 & a_2 & b_2 \\
& b_2 & \ddots & \ddots \\
& & \ddots & \ddots & b_{n-1} \\
& & & b_{n-1} & a_n
\end{bmatrix},\ B = \begin{bmatrix}
1&0&0&\cdots\\
0&a_2 & b_2 \\
0&b_2& a_3 & \ddots & \\
\vdots&\ddots & \ddots & b_{n-1} \\
& & b_{n-1} & a_n
\en... | You have
$$
B=A-D,
$$
where
$$
D=\begin{bmatrix}
a_1-1&b_1&0&0&\cdots&0\\
b_1&0&0&0&\cdots&0 \\
0&0&0&0&\cdots&0 \\
0&0&0&0&\cdots&0 \\
\vdots&\vdots&&\ddots&\ddots&\\
0&0&0&0&\cdots&0 \\
\end{bmatrix}.
$$
Then
$$
C=A^{-1}B=I-A^{-1}D.
$$
Since $D$ has nonzero entries only at the first two columns, we have
$$
A^{-1}D... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2944395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Real Part of the Dilogarithm It is well known that
$$\frac{x-\pi}{2}=-\sum_{k\geq 1}\frac{\sin{kx}}{k}\forall x\in(0,\tau),$$
which gives
$$\frac{x^2}{4}-\frac{\pi x}{2}+\frac{\pi^2}{6}=\sum_{k\geq 1}\frac{\cos(kx)}{k^2}.$$
Note that
$$\textrm{Li}_2(e^{ix})=\sum_{k\geq 1}\frac{\cos(kx)+i\sin(kx)}{k^2}$$
This means tha... | To simplify, you can use the Inverse Tangent Integral $\operatorname{Ti_{2}}$ and the Clausen Functions $\operatorname{C_{n}}$ and $\operatorname{S_{n}}$.
$$
\begin{align*}
\operatorname{Li_{2}}\left( e^{x \cdot i} \right) &= \sum_{k = 1}^{\infty} \left( \frac{\cos\left( k \cdot x \right) + \sin\left( k \cdot x \right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2947873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Simplify the Dot Product in terms of $a$ and $b$ Where $a$ and $b$ are arbitrary vectors
$(a+2b) \cdot (2a-b)$
$$a\cdot(2a-b)+2b\cdot(2a-b) = 2(a\cdot a)-a\cdot b+1(b\cdot a)-2(b\cdot b)$$
$$=2(a)-ab+4ab-2(b)^2$$
$$=2a^2-2b^2$$
$$=2(a^2-b^2)$$
Where did i go wrong in simplifying this?
| $$(\mathbf a+2\mathbf b)\cdot(2\mathbf a-\mathbf b)=2\mathbf a\mathbf a-\mathbf a\mathbf b+4\mathbf b\mathbf a-2\mathbf b\mathbf b=2(|\mathbf a|^2-|\mathbf b|^2)+3\mathbf a\mathbf b$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2948870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Proving $7|13\cdot 6^{n}+8\cdot 13^{n}$ for all natural numbers. I would like to verify whether my proof is correct. The answer sheet used a much more intuitive and logical approach but I think mine is correct also.
To prove: $7|13\cdot 6^{n}+8\cdot 13^{n}$ for all natural numbers
Proof: We proceed by induction and s... | $$13\cdot 6^{n}+8\cdot 13^{n}\equiv (-1)(-1)^n+(1)(-1)^n =0 \text { mod (7) }$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2949124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
Putting the general equation for a circle in terms of x (or y) After about a year of inactivity, I'm back!!
I've been trying to solve a series of math equations by putting them in terms of x, but I've run into a problem with the famous circle equation:
$$a = \sqrt{ ( x - b ) ^ 2 + ( y - c ) ^ 2}$$
where:
a is the radi... | Does this work:
$a = \sqrt{ ( x - b ) ^ 2 + ( y - c ) ^ 2}$
$a^2=(x-b)^2+(y-c)^2$
$(y-c)^2=a^2-(x-b)^2$
$y-c=\pm \sqrt {a^2-(x-b)^2}$
$y=c\pm \sqrt {a^2-(x-b)^2}$
For an equation of the form $x=f(y)$
$a = \sqrt{ ( x - b ) ^ 2 + ( y - c ) ^ 2}$
$a^2=(x-b)^2+(y-c)^2$
$(x-b)^2=a^2-(y-c)^2$
$x-b=\pm\sqrt {a^2-(y-c)^2}$
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2952760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Solve $9^x-2^{x+\frac{1}{2}}=2^{x+\frac{7}{2}}-3^{2x-1}$ $$9^x-2^{x+\frac{1}{2}}=2^{x+\frac{7}{2}}-3^{2x-1}.$$
The equation states solve for $x$.
What I first did was put like bases together.
$$3^{2x}+3^{2x-1}= 2^{x+\frac{7}{2}}+ 2^{x+\frac{1}{2}}.$$
Then I factored $3^{2x}$ and $2^x$
$$3^{2x}(1+\frac{1}{3})=2^x(2^{\fr... | Your last line is wrong.
It should be $$3^{2x-3}=(\sqrt2)^{2x-3},$$ which gives $x=1.5$.
You got:
$$3^{2x}\left(1+\frac{1}{3}\right)=2^x(2^{\frac{7}{2}}+2^{\frac{1}{2}})$$ or
$$3^{2x-1}=2^{x-2}2^{\frac{1}{2}}(1+8)$$ or
$$3^{2x-3}=2^{x-\frac{3}{2}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2955218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Is there an elegant way to determine $Av$ given $Au_1, Au_2$, and $Au_3$ for a $3\times3$ matrix $A$? Let A be a 3x3 matrix such that ${A} \begin{pmatrix} 3 \\ 4 \\ 5 \end{pmatrix} = \begin{pmatrix} 2 \\ 7 \\ -13 \end{pmatrix}, \quad \ {A} \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix} = \begin{pmatrix} -6 \\ 0 \\ 4 \end{pm... | Note that the given conditions can be generalized:
$$\ {A} \begin{pmatrix} 3&4&5 \\ 4&5&-9 \\ 5&6&1 \end{pmatrix}=\begin{pmatrix} 2&-6&3 \\ 7&0&3 \\ -13&4&-11 \end{pmatrix}.$$
Find $A$:
$$A=\begin{pmatrix} 2&-6&3 \\ 7&0&3 \\ -13&4&-11 \end{pmatrix}\cdot \begin{pmatrix} 3&4&5 \\ 4&5&-9 \\ 5&6&1 \end{pmatrix}^{-1}=\begin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2958725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
If $x>0, y>0,x+y=\frac{\pi}{3}$ then maximum value of $\tan x\tan y$
If $x>0, y>0$ and $x+y=\frac{\pi}{3}$, then find the maximum value of $\tan x\tan y$
My Attempt
$x>0, y>0, x+y=\frac{\pi}{3}\implies x, y$ in $1^\text{st}$ quadrant.
$\tan x, \tan y>0, \tan(x+y)=\sqrt{3}, \tan x\tan y>0$
$$
\tan x+\tan y\geq2\sqrt{\... | For $-\dfrac{\pi}{6} < t < \dfrac{\pi}{6}$, let $x = \dfrac{\pi}{6} - t \ $ and $ \ y = \dfrac{\pi}{6} + t$.
Then $x > 0$, $y>0$ and $x + y = \dfrac{\pi}{3}$.
Let
\begin{align}
f(t)
&= \tan(x) \tan(y) \\
&= \tan\left(\dfrac{\pi}{6} - t \right) \tan \left(\dfrac{\pi}{6} + t \right) \\
&= \dfrac{1-\sqrt 3 \t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2958959",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Find exhaustive range of $k$ such that $f(x)=\frac{x-1}{k-x^2}$ never belongs to $\left[-1 \:\: \frac{-1}{3}\right]$ Find exhaustive range of $k$ such that $$f(x)=\frac{x-1}{k-x^2}$$ never belongs to $\left[-1 \:\: \frac{-1}{3}\right]$
My try:
Letting $$y=\frac{x-1}{k-x^2}$$ we get
$$yx^2+x-(1+ky)=0$$ and since $x\in \... | $f(x)=\frac{x-1}{k-x^2}$ never belongs to $[-1,-\frac{1}{3}]$
$\iff$ For every $y$ satisfying $-1\le y\le -\frac 13$, there are no $x$ such that $y=\frac{x-1}{k-x^2}$
$\iff$ For every $y$ satisfying $-1\le y\le -\frac 13$, there are no $x$ such that $y(k-x^2)=x-1$ and $k-x^2\not=0$
(If $k-x^2=0$, then $x=1$ implying $k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2960218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Prove that $\frac{1+\sin\theta + i\cos\theta}{1+\sin\theta-i\cos\theta}=\sin\theta+i\cos\theta.$ Prove that $$\frac{1+\sin\theta + i\cos\theta}{1+\sin\theta-i\cos\theta}=\sin\theta+i\cos\theta.$$
Hence, show that $$(1+\sin\frac{\pi}{5}+i\cos\frac{\pi}{5})^5+i(1+\sin\frac{\pi}{5}-i\cos\frac{\pi}{5})^5=0.$$
In the first ... | Hint:
$$\frac{1+sin\theta + icos\theta}{1+sin\theta-icos\theta}=sin\theta+icos\theta \iff 1+sin\theta + icos\theta =\underbrace{(1+sin\theta-icos\theta)(sin\theta+icos\theta)}_{R}$$
so calculate the right side and...
$$ R = sin\theta+icos\theta + \underbrace{sin^2\theta-i^2cos ^2\theta}_{=1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2961689",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Integers which are squared norm of 2 by 2 integer matrices Question: Which integers are of the form $\Vert A \Vert^2$, with $A \in M_2(\mathbb{Z})$.
The code below provides the first such integers: $0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26$.
By searching this sequence on OEIS, we find: "Numbers that are the ... | Yes. If $A$ is a $2\times2$ integer matrix such that $n=\|A\|^2$ is an integer, $n$ must be the sum of two integer squares. Conversely, if $n$ is the sum of two integer squares, then $n=\|A\|^2$ for some $2\times2$ integer matrix $A$.
Proof. Suppose $A$ is a $2\times2$ integer matrix such that $n=\|A\|^2$ is an integer... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2963069",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 1,
"answer_id": 0
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Proving a binomial coefficient inequality per induction $\left(\begin{array}{c}2n\\ n+k\end{array}\right) < \left(\begin{array}{c}2n\\ n\end{array}\right), n,k \in \mathbb{N}, 1\leq k \leq n$
Proving this by induction on $k$ is no problem. However, I want to prove this by induction on $n$, but I am stuck because of a t... | Choose $c$ s.t. $\left(\begin{array}{c}2n\\ n+k\end{array}\right) < c \times \left(\begin{array}{c}2n\\ n\end{array}\right) < \left(\begin{array}{c}2n\\ n\end{array}\right)$ . $c = \frac{(n+1-k)(n+1+k)}{(n+1)(n+1)} $. The inductive step (first inequality) works. The second inequality is true since $ \frac{(n+1-k)(n+1+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2963266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Using the sequential definition of a limit to show $\lim_{x\to 1} \frac{x^2 - 1}{\sqrt{x} - 1} = 4.$ Very recently, I posted this thread: Using the sequential definition of a limit to show $\lim_{x\to 0} \frac{x^2}{x} = 0.$ My solution for that proof was correct, but now I'm having trouble showing that $\lim_{x\to 1} \... | Hint
Use that
$$t^4-1=(t-1)(1+t+t^2+t^3)$$
with $t=\sqrt x.$
Edit
You got
$$\left|\frac{x_{n}^{2} - 1}{\sqrt{x_{n}} - 1} - 4\right|
= \left|1 + \sqrt{x_{n}} + x_{n} + \sqrt{x_{n}^{3}} - 4\right|.$$
Assume that $1-\delta<x_n\le 1.$ Then we have
$$1 + \sqrt{x_{n}} + x_{n} + \sqrt{x_{n}^{3}}>1+\sqrt{1-\delta}+1-\delta+\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2965202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Help with this proof by induction with inequalities. Show that mathematical induction can be used to
prove the stronger inequality $\frac{1}{2}\cdot ...\cdot \frac{2n-1}{2n} < \frac{1}{\sqrt{3n + 1}}$ for all integers greater than 1, which, together with a verification for the case where n = 1, establishes the weaker i... | Continuing where you left. You need to show $\large{\frac{1}{\sqrt{3n+1}}\cdot \frac{2n+1}{2n+2}<\frac{1}{\sqrt{3n+4}}}$, as square roots defined as positive numbers we can multiply both side by $\sqrt{3n+1},$ and this implies $$\frac{2n+1}{2n+2}< \frac{\sqrt{3n+1}}{\sqrt{3n+4}}\implies (2n+1)\sqrt{3n+4}<(2n+2)\sqrt{3n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2965485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Finding minimum value of $ \frac{x^2 +y^2}{y} $
Finding the minimum value of $\displaystyle \frac{x^2 +y^2}{y}.$ where $x,y$ are
real
numbers satisfying $7x^2 + 3xy + 3y^2 = 1$
Try: Equation $7x^2+3xy+3y^2=1$ represent Ellipse
with center is at origin.
So substitute $x=r\cos \alpha $ and $y=r\sin \alpha$
in $7... | The problem has no solution (assuming you are looking for a (global) minimum rather than just a local minimum). As we approach the point $(1/\sqrt{7},0)$ along the constraint curve from below the $x$-axis, the value of the function goes to $-\infty$:
The points
$$\left(\frac{\sqrt{28-75{y}^{2}}-3y}{14},y\right)$$
sati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2965865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
What is the smallest positive integer $n$ such that $(a+1)^{7^n} = (a+1)$? In $\mathbb{F} = \mathbb{F}_7[X]/(X^2+1)$ let $a$ be the class of $X$. What is the smallest positive integer $n$ such that $(a+1)^{7^n} = (a+1)$ ?
I know that the answer is $2$ but I don't see how it works. Can someone explain the logic to me? B... | The question can be rephrased as
What is the smallest positive integer $n$ such that $(a+1)^{7n-1} = 1$?
Let $m$ be the order of $a+1$ in the multiplicative group $\Bbb{F}^\times$, which has order $48$.
Then $m$ divides $\gcd(7n-1,48)$. Let's see the possibilities:
$$
\begin{array}{r|rrrrrrr}
n & 1 & 2 & 3 & 4 & 5 & ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2969252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Evaluate $\int\limits_{|z|=1}\frac{\sin{1/z}}{(z-2)^2}dz.$ This is a question from an old exam. The two integrals are seemingly similar, but the first one seems quite tedious compared to the other one. It seems, according to the prof solution, that the hard part in the first integral is computing the residue at $z=0.$
... | Re $(a)$ and series, given
$$\sin{z}=z-\frac{z^3}{3!}+\frac{z^5}{5!}-\frac{z^7}{7!}+\frac{z^9}{9!}+...=
\sum\limits_{k=0}\frac{(-1)^k}{(2k+1)!}z^{2k+1} \tag{1}$$
we have
$$\sin{\frac{1}{z}}=\sum\limits_{k=0}\frac{(-1)^k}{(2k+1)!}\frac{1}{z^{2k+1}}$$
then
$$\int\limits_{|z|=1}\frac{\sin{(1/z)}}{(z-2)^2} dz=\sum\limits_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2969634",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Find Derivative of $y=\tan^{-1}\left(\sqrt{\frac{x+1}{x-1}}\right)$ for $|x|>1$?
Question. If $y=\tan^{-1}\left(\sqrt{\frac{x+1}{x-1}}\right)$ for $|x|>1$ then $\frac{d}{dx}=?$
Answer: $\displaystyle\frac{dy}{dx}=-\frac{1}{2|x|\sqrt{x^2-1}}$
My 1st attempt- I followed the simple method and started by taking darivati... | You start right:
\begin{align}
\frac{dy}{dx}
&=\frac{1}{1+\left(\sqrt{\dfrac{x+1}{x-1}}\right)^2}
\frac{1}{2\sqrt{\dfrac{x+1}{x-1}}}
\frac{(x-1)-(x+1)}{(x-1)^2}\\
&=\frac{1}{2}\frac{x-1}{(x-1)+(x+1)}
\sqrt{\dfrac{x-1}{x+1}}
\frac{-2}{(x-1)^2}
\end{align}
but then make a decisive error in splitting the square root... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2970280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 1
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Verify the proof that $x_n = \ln^2(n+1) - \ln^2n$ is a bounded sequence.
Let $n\ \in \mathbb N$ and:
$$
x_n = \ln^2(n+1) - \ln^2n
$$
Prove that $x_n$ is a bounded sequence.
I've taken the following steps. Consider $x_n$
$$
\begin{align}
x_n &= \ln^2(n+1) - \ln^2n = \\
&= (\ln(n+1) + \ln n)(\ln (n+1) - \ln n) = \... | Lagrange's theorem provides a one-liner:
$$ \log^2(n+1)-\log^2(n) = \frac{d}{dx}\left.\log^2(x)\right|_{x=\xi\in(n,n+1)},\quad \frac{2\log\xi}{\xi}\to 0\text{ as }n\to +\infty. $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2972286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
find the sum to n term of $\frac{1}{1\cdot2\cdot3} + \frac{3}{2\cdot3\cdot4} + \frac{5}{3\cdot4\cdot5} + \frac{7}{4\cdot5\cdot6 } + ... $ $$\frac{1}{1\cdot2\cdot3} + \frac{3}{2\cdot3\cdot4} + \frac{5}{3\cdot4\cdot5} + \frac{7}{4\cdot5\cdot6 } + ... $$
$$=\sum \limits_{k=1}^{n} \frac{2k-1}{k\cdot(k+1)\cdot(k+2)}$$ $$= ... | HINT:
Note that we have
$$\begin{align}
\frac{2k-1}{k(k+1)(k+2)}&=\color{blue}{\frac{3}{k+1}}-\frac{5/2}{k+2}-\frac{1/2}{k}\\\\
&=\color{blue}{\frac12}\left(\color{blue}{\frac{1}{k+1}}-\frac1k\right)+\color{blue}{\frac52}\left(\color{blue}{\frac{1}{k+1}}-\frac{1}{k+2}\right)
\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2972505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 0
} |
Reflection of point in barycentric coordinates I have a triangle $ABC$ and a point $P$ with barycentric coordinates ($\alpha, \beta, \gamma)$ that I want to reflect about the sides $a,b$ and $c$.
Calculating the general expression for a displacement vector perpendicular to $c$ and then using $|PB|=|P'B|$, I got
$$P'=\l... | Let's try some brute force with a special case. Assign Cartesian coordinates
$$A := (0,0) \qquad B := (c,0) \qquad C := (b\cos A, b\sin A) \tag{1}$$
and define $P$ with barycentric coordinates $(\alpha, \beta, \gamma)$, so that
$$P = \frac{\alpha A + \beta B + \gamma C}{\alpha + \beta + \gamma} = \frac{(\beta c+\gamma ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2972761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Divisibility 1,2,3,4,5,6,7,8,9,&10 Tried:
Seems the ten-digit number ends with $240$ or $640$ or $840$ (Is not true, there are more ways the number could end)
$8325971640,$
$8365971240,$
$8317956240,$
$8291357640,$
$8325971640,$
$8235971640,$
$1357689240,$
$1283579640,$
$1783659240,$
$1563729840,$
$1763529840,$
$165372... | Suppose the number is of form $N=jihgfedcba$. We may write:
$a+b+c+d+e+f+g+h+i+j+(10-1)b+(10^2-1)c+(10^3-1)d+(10^4-1)e+(10^5-1)f+(10^6-1)g+(10^7-1)h+(10^8-1)i+(10^9-1)j$ | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2972869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Trigonometric Identities: Given $\cot(b)=-2$ find $\sin(4b)$ and $\cos(4b)$
Given $\cot(b)=-2$ find $\sin(4b)$ and $\cos(4b)$
I found $\sin(4b)$ as $-24/25$ but I always get a very bad answer for $\cos(4b)$
Any hints?
edit:
I made $sin(4b)$ into a expanded form.
$4sin(b)cos^3(b)-4sin^3(b)cos(b)$
And then I made a tri... | The result $353/25$ is surely wrong, as a cosine cannot be $>1$.
You're complicating your own life! ;-)
There are standard formulas:
\begin{align}
\sin 2x&=\frac{2\tan x}{1+\tan^2x}=\frac{2\cot x}{\cot^2x+1}
\\[4px]
\cos 2x&=\frac{1-\tan^2x}{1+\tan^2x}=\frac{\cot^2x-1}{\cot^2x+1}
\end{align}
Thus
$$
\sin 2b=\frac{-4}{4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2973963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Inequality sign changing when multiplying with a unknown variable? This is the inequality:
$$\frac{(2x-1)}{(3-5x)} < 2$$
Next step is:
$$\frac{(2x-1)}{(3-5x)} \cdot (3-5x) < 2\cdot (3-5x)$$
Now is the sign $<$ or $>$?
The end solution is $x < \frac{7}{12}.$
| In general, when multiplying an inequality by an unknown quantity, we can't tell whether the sign changes or not. To solve the problem, then, we must consider both cases, i.e. whether the unknown quantity is positive or negative. In this example, we would do the following:
CASE 1: $3-5x > 0$
\begin{align*}
\frac{2x-1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2974876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Proving $\sum\limits_{\text{cyc}} \frac{a}{b^2+c^2+d^2} \geq \frac{3\sqrt{3}}{2}\frac{1}{\sqrt{a^2+b^2+c^2+d^2}}$ for $a, b, c, d >0$
Let $a, b, c, d > 0$. Prove that
$$\frac{a}{b^2+c^2+d^2}+\frac{b}{a^2+c^2+d^2}+\frac{c}{a^2+b^2+d^2}+\frac{d}{a^2+b^2+c^2}$$
$$\geq\frac{3\sqrt{3}}{2}\frac{1}{\sqrt{a^2+b^2+c^2+d^2}}.$$... | Since the desired inequality is homogeneous, assume that $a^2 + b^2 + c^2 + d^2 = 1$. The desired inequality becomes
$$\sum_{\mathrm{cyc}}\frac{a}{1 - a^2} \ge \frac{3\sqrt 3}{2}.$$
Note that
\begin{align*}
\frac{a}{1 - a^2} - \frac{3\sqrt 3}{2}a^2
&= \frac{a}{2(1 - a^2)}
[2 - 3\sqrt 3\, a(1 - a^2)]\\
&= \frac{a}{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2975426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Formal proof of the sequence involving double factorials $x_n = \frac{(2n)!!}{(2n-1)!!}$ is not bounded. I'm trying to prove the following:
Let $n\in \mathbb N$ and
$$
x_n = \frac{(2n)!!}{(2n-1)!!}
$$
Where:
$$
(2n)!! = 2\cdot 4 \cdot 6 \cdot \dots \cdot 2n \\
(2n - 1)!! = 1\cdot 3 \cdot 5 \cdot \dots \cdot (... | Without integrals or derivatives:
$$ x_n=\frac{(2n)!!}{(2n-1)!!}= \prod_{k=0}^{n-1}\left(1+\frac{1}{2k+1}\right)=\prod_{k=1}^{n}\left(1-\frac{1}{2k}\right)^{-1}$$
implies
$$ x_n^2 = \prod_{k=1}^{n}\left(1-\frac{1}{k}+\frac{1}{4k^2}\right)^{-1}=\frac{1}{4}\prod_{k=2}^{n}\left(1-\frac{1}{k}\right)^{-1}\prod_{k=2}^{n}\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2976028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Power Series (Cambridge Tripos 1900) If
$$a/(a+bz+cz^2)=1+p_1z+p_2z^2
+\dots$$
then
$$1+p_1^2z+p_2^2z^2+\dots=\frac{a+cz}{a-cz}\frac{a^2}{
a^2-(b^2-2ac)z+c^2z^2}$$
A tricky problem from G.H.Hardy's "A Course in Pure Mathematics", or maybe I'm missing the obvious.Any help will be greatly appreciated.
| If $\alpha,\beta$ are roots of $az^2+bz+c$ then we have $(a+bz+cz^2)/a=(1-\alpha z)(1-\beta z)$, so
$$
p_n=\alpha^n+\alpha^{n-1}\beta+\dots+\alpha\beta^{n-1}+\beta^n=\frac{\alpha^{n+1}-\beta^{n+1}}{\alpha-\beta}
$$
Thus
$$
p_n^2=\frac{1}{(\alpha-\beta)^2}\left[\alpha^{2n+2}-2\alpha^{n+1}\beta^{n+1}+\beta^{2n+2}\right]
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2977675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Numerical method to find the roots of the derivative of $y = (x^3 - 4x) \cdot \frac{\sqrt{13+4x^2}}{2}$ (Newton's Method) I am working on an Optimization problem and I would like to suggest finding the solution (i.e. the extrema) of the function below using two methods: the first one being calculating the derivative an... | We want to find the extrema of the function
$$\begin{align*}
y(x) = (x^3 - 4x) \cdot \frac{\sqrt{13+4x^2}}{2}
\end{align*}$$
One approach is to use Newton's Method on $y'(x)$, where
$$y'(x) = \frac{16 x^4+7 x^2-52}{2 \sqrt{4 x^2+13}}$$
We start by plotting $y(x)$ and see two extrema to try and find using Newton's
The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2978945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Prove that $10|n+3n^3+7n^7+9n^9$ Prove that $10|n+3n^3+7n^7+9n^9$ for every $n\in \mathbb N$
Only what i see that 10=5*2 and both number is free numbers, and if I show that $5|n+3n^3+7n^7+9n^9$ and $2|n+3n^3+7n37+9n^9$ that I prove, since 5 and 2 is free numbers i can use Fermat's little theorem such that $5|n^5-n$ an... | Hint:
$$n+3n^3+7n^7+9n^9\equiv n+3n^3-3n^7-n^9\equiv -(n^4-1)(n)(n^4+3n^2+1)$$
mod 10, so prove
$$2|n(n^4-1)$$
and
$$5|n^4-1$$
when $n\neq5k$ and $5|n$ when $n=5k$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2979515",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 3
} |
$2 < (1 + \frac{1}{n})^n < 3$ for all $n > 2$ Show by induction that $2 < (1 + \frac{1}{n})^n < 3$ for all natural n > 2.
*
*Induction base. For $n = 3$ the inequality is obviously true.
*Assume that for $n = k$ inequality is true, than i can prove $(1 + \frac{1}{n+1})^{n +1} < 3$.
$$\frac{(n + 1)^{n}}{n^{n}} < 3 ... | HINT
What about Bernoulli's inequality?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2980483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Prove the inequality: $\left(\frac{a}{a+2b}\right)^2+\left(\frac{b}{b+2c}\right)^2+\left(\frac{c}{c+2a}\right)^2 \geq \frac{1}{3}$
Let $a,b,c$ are all positive real numbers. Prove
$$\left(\frac{a}{a+2b}\right)^2+\left(\frac{b}{b+2c}\right)^2+\left(\frac{c}{c+2a}\right)^2 \geq \frac{1}{3}$$
Can anyone give a hint?
| By Cauchy-Schwarz,
$$\sum \limits_{\text{cyc}} \left ( \dfrac{a}{a+2b} \right )^2 \geq \dfrac{1}{3} \left(\sum \limits_{\text{cyc}} \left ( \dfrac{a}{a+2b} \right )\right)^2 = \dfrac{1}{3}\left( \sum \limits_{\text{cyc}} \left ( \dfrac{a^2}{a^2+2ab} \right) \right)^2 \geq \dfrac{1}{3} \left ( \dfrac{(a+b+c)^2}{(a+b+c)^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2983036",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What mistake am I making when trying to do REF? I hope this kinda of question is allowed. I know that you're all uh.. very.. particular, for a lack of a better term, about the kinds of questions you get. I am just practicing a the most basic of reducing a matrix into row echelon form. I am getting a similar, but differ... | In the first step we should have
$$\begin{pmatrix}
1 & 0 & -3 & 4 \\
0 & 1 & 1 & -3 \\
\color{red}2 & 2 & -5 & 2
\end{pmatrix}$$
moreover we don't need to exchange rows at first, we can proceed as follow
$$\begin{pmatrix}
2 & 2 & -5 & 2 \\
1 & 1 & -2 & 1 \\
1 & 0 & -3 & 4
\end{pmatrix} \stackrel{R3-R2}\to \begin{pmat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2983349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Find the value of $(\frac{\alpha}{\alpha +1})^3+(\frac{\beta}{\beta +1})^3+(\frac{\gamma}{\gamma +1})^3$ Find the value of
$(\frac{\alpha}{\alpha +1})^3+(\frac{\beta}{\beta +1})^3+(\frac{\gamma}{\gamma +1})^3$ where $\alpha,\beta,\gamma$ are roots of the equation $x^3+2x^2+3x+3=0$.
I tried to use the formula which is... | Substituting $x=\frac y{1-y}$, which is the inverse of $y=\frac x{x+1}$, yields
$$y^3-5y^2+6y-3=0$$
and $a,b,c=\frac\alpha{1+\alpha},\frac\beta{1+\beta},\frac\gamma{1+\gamma}$ are its roots. Then the desired expression is
$$a^3+b^3+c^3=(a+b+c)^3-3(a+b+c)(ab+bc+ca)+3abc$$
and we can extract the values of the symmetric ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2983790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
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Is $x(1 - 2x) \le \frac{1}{8}$ and further, is $x(1 - ax) \le \frac{1}{4a}$ It is clear that $x(1-x) \le \frac{1}{4}$
Does it likewise follow that $x(1-2x) \le \frac{1}{8}$?
Here's my reasoning:
(1) For $x < \frac{1}{4}$, $x(1-2x) < \frac{1}{8}$
(2) For $\frac{1}{4} < x < \frac{1}{2}$, $x(1-2x) < \frac{1}{8}$
(3) Fo... | The first it's
$$16x^2-8x+1\geq0$$ or
$$(4x-1)^2\geq0.$$
The second for $a>0$ it's
$$4a^2x^2-4ax+1\geq0$$ or $$(2ax-1)^2\geq0.$$
For $a<0$ the second inequality is reversed.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2983897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 0
} |
Number of solutions in degree four Find number of postive integra solutions of the equation
$ x^4+ 4y^4 + 16z^4 +64= 32xyz$.
I could just proceed till that x cant be odd.
| There are no integer solutions.
As you remark, $x$ must be even. Write $x=2x_1$. Then we see $$16x_1^4+4y^4+16z^4+64=64x_1yz\implies 16\,|\,4y^4\implies y=2y_1$$
Continuing we get $$4x_1^4+16y_1^4+4z^4+16=16x_1y_1z\implies x_1^4+4y_1^4+z^4+4=4x_1y_1z$$
Thus, $x_1\equiv z\pmod 2$. If they were both odd, we'd have $x_1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2984252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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How to solve this radical expression I've been trying to solve this expression for at least two hours now... And I always get stuck towards the end, I don't know what I'm missing.
$\frac 1{xy} \times (\sqrt{xy} - \frac{xy}{x-\sqrt{xy}})\times (\sqrt{xy} + \frac{xy}{x+\sqrt{xy}})$
My first step was to rationalize the fr... | We have
$$\frac 1{xy} \times (\sqrt{xy} - \frac{xy}{x-\sqrt{xy}})\times (\sqrt{xy} + \frac{xy}{x+\sqrt{xy}})
=\frac 1{xy} \times \left(xy - \frac{x^2y^2}{x^2-xy}+\frac{xy\sqrt{xy}}{x+\sqrt{xy}}-\frac{xy\sqrt{xy}}{x-\sqrt{xy}}\right)=$$
$$=1 - \frac{xy}{x^2-xy}+\frac{\sqrt{xy}}{x+\sqrt{xy}}-\frac{\sqrt{xy}}{x-\sqrt{xy}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2986340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find the final tableau knowing the optimal solution
Consider the following linear program $$\displaystyle \max z=5x_1+2x_2+3x_3\\ s.t. x_1+5x_2+2x_3\le b_1\\ x_1-5x_2-6x_3\le b_2 \\
x_1,x_1,x_3\ge0$$
If the optimal solution is reached at $x_1=30,x_5=10,$ write directly the complete optimal tableau, whitout executing t... | From the given optimal solution $z(x_1,x_2,x_3,x_4,x_5)=z(30,0,0,0,10)=150$, you can find:
$$x_1+5x_2+2x_3+x_4=b_1=\color{red}{30},\\ x_1-5x_2-6x_3+x_5=b_2=\color{red}{40},
$$
where $x_4$ and $x_5$ are the slack variables.
Since in the optimal table only the slack variable $x_4$ is replaced with $x_1$, then the inter... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2986579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
ّFind $x$ such that $ \frac{1}{x^2} + \frac{1}{(3-x)^2} = \frac{104}{25}$.
ّFind $x$ such that
$$ \frac{1}{x^2} + \frac{1}{(3-x)^2} = \frac{104}{25}\,.$$
My attempt:
After clearing the denominators, I obtain this quartic equation
$$104 x^{4} -624 x^{3} +886 x^{2} +150x-225=0.$$
I don't know how to proceed from here.
| Now, you can make the following.
Easy to see that $\frac{1}{2}$ and $\frac{5}{2}$ they are roots of the equation,
which gives a factor $$(2x-1)(2x-5)=4x^2-12x+5$$ and
$$104x^4-624x^3+886x^2+150x-225=$$
$$=104x^2-312x^3+130x^2-312x^3+936x^2-390x-180x^2+540x-225=$$
$$=26x^2(4x^2-12x+5)-78(4x^2-12x+5)-45(4x^2-12x+5)=$$
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2988342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 3
} |
What are the coefficients in the expansion of $(x+y)(x+2y) \cdots (x+ny)$? Are the numbers appearing as coefficients in the following sequence of polynomials known? Is there a known recurrence relation to compute them?
\begin{align*}
(x+y) &= x+y \\
(x+y)(x+2y) &= x^2+3yx+2y^2 \\
(x+y)(x+2y)(x+3y) &= x^3+6yx^2+11... | The On-Line Encyclopedia of Integer Sequences® is a
searchable database of many integer sequences, and (as @Kevin also said in a comment) can be a valuable
tool to identify a particular sequence. Searching for
$$
1, 1, 1, 3, 2, 1, 6, 11, 6, 1, 10, 35, 50, 24
$$
returns as the top result
A094638 Triangle read by ro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2988624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 1,
"answer_id": 0
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Problem in reducing a expression In my previous question, I was able to solve it using Weierstrass substitution as the answers pointed out.
But I was keen to know if my expression can be reduced to the given answer.
Desmos proves that the two integrals are indeed equal, here's the proof.
Now how do I reduce this expres... | Let's denote $$I_1=\int_{0}^{\sqrt{2}-1}\frac{2}{(1+t)\bigg(\sqrt{(1+t)+\sqrt{(1+t)^2-1}}+\sqrt{(1+t)-\sqrt{(1+t)^2-1}}\bigg)}dt$$ and $$I_2=\int_{0}^{\sqrt{2}-1}\frac{4t}{(1+t^2)\sqrt{1-t^2}}dt$$
First let's simplify the denominator in the first integral. Since both terms are positive we are able to do the following t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2989841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving continuity in the origin $\frac{xy}{\sqrt{x^2 + y^2}}$ Let $g: \mathbb{R^2} \to \mathbb{R}$.
How can I prove that $g$ is continuous in its origin, but not totally differentiable?
If I take
$$g(\frac{1}{n},\frac{1}{n}) = \frac{1}{n\sqrt{2}} \to 0 \text{ for } n \to \infty $$
Or rather:
$$|x,y| \leq \frac{1}{2... | Note that $x^2+y^2\ge 2|xy|$. Hence, we have
$$\left|\frac{xy}{\sqrt{x^2+y^2}}\right| \le \frac{\sqrt{|xy|}}{\sqrt 2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2990614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Polynomial division with remainder
If the polynomial
$$x^4-x^3+ax^2+bx+c$$
divided by the polynomial
$$x^3+2x^2-3x+1$$
gives the remainder $$3x^2-2x+1$$
then how much is (a+b)c?
So what I know, and how I solved these problems before, I can write this down like this:
$x^4-x^3+ax^2+bx+c=Q(x)(x^3+2x^2-3x+1)+3x... | Hint: Compute the remainder of $x^4-x^3$ when divided by $x^3+2x^2-3x+1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2993694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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$M$ is a point in an equaliateral $ABC$ of area $S$. $S'$ is the area of the triangle with sides $MA,MB,MC$. Prove that $S'\leq \frac{1}{3}S$.
$M$ is a point in an equilateral triangle $ABC$ with the area $S$. Prove that $MA, MB, MC$ are the lengths of three sides of a triangle which has area $$S'\leq \frac{1}{3}S$$
|
Let $N$ be the rotation of $M$ about $C$ by an angle of $60^\circ$, then $\triangle AMN$ has three sides equal to $MA,MB, MC$ respectively. Now
$$\begin{aligned}
P &= \frac12 MN\cdot MA\cdot \sin(\angle AMN)\\
&= \frac12 MC \cdot MA\cdot\sin(\angle AMC - 60)\\
&= \frac12 MC \cdot MA\cdot (\sin(\angle AMC)\cos 60 - \co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2993965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Prove $\int^{\infty}_{-\infty}\frac{1}{(x^2+x+1)^3}dx=\frac{4\pi}{3\sqrt3}$. I have no idea how to do this question.
I'm given $\int^{\infty}_{-\infty}\frac{1}{x^2+2ax+b^2}dx=\frac{\pi}{\sqrt{b^2-a^2}}$ if $b>|a|$ and I'm asked to prove $\int^{\infty}_{-\infty}\frac{1}{(x^2+x+1)^3}dx=\frac{4\pi}{3\sqrt3}$.
What I've tr... | For any $A>0$ we have $\int_{-\infty}^{+\infty}\frac{dz}{z^2+A} = \frac{\pi}{\sqrt{A}}$, hence by applying $\frac{d^2}{dA^2}$ to both sides we get
$$ \int_{-\infty}^{+\infty}\frac{dz}{(z^2+A)^3}=\frac{3\pi}{8A^2\sqrt{A}}.\tag{1} $$
On the other hand
$$ \int_{-\infty}^{+\infty}\frac{dx}{(x^2+x+1)^3}\stackrel{x\mapsto z-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2994751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Show that $\det\left[\begin{smallmatrix}1&\cos a&\cos b\\ \cos a&1&\cos(a+b) \\ \cos b&\cos(a+b)&1 \end{smallmatrix}\right]=0$ I am unable to show - without expanding, by using determinant properties - that
$$\det\begin{bmatrix} 1 &\cos a &\cos b\\
\cos a &1 ... | Here we go: directly
$$\begin{align}\begin{vmatrix} 1 &\cos a &\cos b\\
\cos a &1 &\cos(a+b) \\
\cos b &\cos(a+b) &1
\end{vmatrix}&=\\&=
[1+2\cos a\cos b\cos(a+b)]-[\cos^2b+\cos^2a+\cos^2(a+b)]\\&=
1+\cos(a+b)\cdot [2\co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2994980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
A question about the accumulation points of a weird sequence. Recently I came across a strange problem asking me to find the accumulation points of the sequence: $a_n$=$\sqrt[3]{n}\cdot\sin\sqrt{n}$
I really don't know how to write in detail, although I'm sort of sure that the accumulation points should be all $\mathbb... | Fix any $x \in \mathbb{R}$ and $\varepsilon > 0$. WLOG, assume $\varepsilon < |x|/2$. We want to show that there is some $n$ such that $|a_n - x| < \varepsilon$.
*
*We intend to describe the ''distribution'' of $\sqrt{n}$. The distance between two adjacent points is $$\sqrt{n+1} - \sqrt{n} = \frac{1}{\sqrt{n} + \sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2996603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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What am I doing wrong finding the limit of $\left(\frac{3x^2-x+1}{2x^2+x+1}\right)^\left(\frac{x^3}{1-x}\right)$? $$\lim_{x\to\infty}\left(\frac{3x^2-x+1}{2x^2+x+1}\right)^\left(\frac{x^3}{1-x}\right)$$
Divide by $x^2$, get
$$\lim_{x\to\infty}(1,5)^\infty=\infty$$
The answer in the book is $0$.
I've also tried substitu... | Note that
$$\frac{3x^2-x+1}{2x^2+x+1} \to \frac32$$
but
$$\frac{x^3}{1-x}=-\frac{x^3}{x-1}\to -\infty$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2998322",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $a$, $b$, $c$ are positive real numbers such that $a^2 + b^2 + c^2 = 27$. Find the least value of $a^3 + b^3 + c^3$? If $a$, $b$, $c$ are positive real numbers such that $a^2 + b^2 + c^2 = 27$. Find the least value of $a^3 + b^3 + c^3$ ?
I tried with Tchebyshev inequality on sets $\{a, b, c\}$ and $\{a^2, b^2 , c^2\... | From the power mean inequality we know that
$$\sqrt[q]{\sum_{i=1}^nw_ix_i^q} \geq \sqrt[p]{\sum_{i=1}^nw_ix_i^p} \quad \forall \ \ p <q$$
where $ \sum_{i=1}^n w_i=1 $
In your case $p=2, q=3$ and $n=3$. Hence,
$$\left(\dfrac{a^3+b^3+c^3}{3}\right)^{1/3}\ge \left(\dfrac{a^2+b^2+c^2}{3}\right)^{1/2}=3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3001046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Evaluating $\int_0^1\frac{\ln(1+x-x^2)}xdx$ without using polylogarithms.
Evaluate $$I=\int_0^1\frac{\ln(1+x-x^2)}xdx$$ without using polylogarithm functions.
This integral can be easily solved by factorizing $1+x-x^2$ and using the values of dilogarithm at some special points.
The motivation of writing this post i... | $$\begin{aligned}
I&=\int_0^1\frac{\ln(1+x-x^2)}x\mathrm{d}x\\
&\overset{(1)}{=}\int_0^1\sum_{n=1}^\infty\frac{(-1)^{n-1}(x-x^2)^n}{nx}\mathrm{d}x\\
&\overset{(2)}{=}\sum_{n=1}^\infty\frac{(-1)^{n-1}}n\int_0^1x^{n-1}(1-x)^n\mathrm{d}x\\
&\overset{(3)}{=}\sum_{n=1}^\infty\frac{(-1)^{n-1}}n\frac{(n-1)!n!}{(2n)!}\\
&=\sum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3002309",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 0
} |
How do I simplify this expression? How do I simplify:
$$d=\frac{-(x+\alpha x)^2+(y+\alpha y)^2+2+x^2-y^2-2}{\sqrt{\alpha^2 x^2+\alpha^2 y^2}}$$
If simplification is possible, it should be possible with elementary algebra, but I'm completely lost as to how to go about it.
What I've done so far:
$$d=\frac{-(x+\alpha x)^... | I believe your third equality is incorrect. Numerator distribution won't lead to any terms with x and y to the first power.
Then when you squared d below, you also have additional squares added to a in denominator where there were none as well as a double square for the entire numerator and denominator - essentially... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3003860",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Binomial Theorem with Three Terms $(x^2 + 2 + \frac{1}{x} )^7$
Find the coefficient of $x^8$
Ive tried to combine the $x$ terms and then use the general term of the binomial theorem twice but this does seem to be working.
Does anyone have a method of solving this questions and others similar efficiently?
Thanks.
| Let $$R(x)= \left(x^2+2+{1\over x}\right)^7$$ then we need to find a coeficient at $x^{15}$ for$$x^7R(x)= (x^3+2x+1)^7 $$ $$= \sum _{k=0}^7 {7\choose k}x^{21-3k}(2x+1)^k$$
Clearly if $21-3k\geq 16$ there is no term with $x^{15}$ so $21-3k\leq 15$ so $k\geq 2$.
Also if $21-3k\leq 7$ we have no term with $x^{15}$ so $2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3004752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 0
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Double Infinite sum of $1/n^2$ I am trying to use an identity we showed on our homework:
$$ \sum_{-\infty}^{\infty} \frac{1}{(n+a)^2} = \frac{\pi^2}{\sin^2(\pi a)} $$
to show that $$ \sum_{1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}.$$
I have broken the first double infinite sum into the sum from $-\infty$ to $1$ p... | Your idea of using Taylor expansions is good.
Let us compose the series around $a=0$
$$\sin(\pi a)=\pi a-\frac{\pi ^3 a^3}{6}+\frac{\pi ^5 a^5}{120}+O\left(a^7\right)$$
$$\sin^2(\pi a)=\pi ^2 a^2-\frac{\pi ^4 a^4}{3}+\frac{2 \pi ^6 a^6}{45}+O\left(a^8\right)$$
$$\frac{\pi^2}{\sin^2(\pi a)}=\frac{\pi^2}{\pi ^2 a^2-\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3005902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Does $\left\lfloor\frac{x^2+x}{i}\right\rfloor - \left\lfloor\frac{x^2}{i}\right\rfloor = \left\lfloor\frac{x}{i}\right\rfloor$? Does $\left\lfloor\dfrac{x^2+x}{i}\right\rfloor - \left\lfloor\dfrac{x^2}{i}\right\rfloor = \left\lfloor\dfrac{x}{i}\right\rfloor$?
where $i \le x^2$ and $x$ are any positive integer.
Intuiti... | Consider x = 7 and i = 5.
you get
$\left\lfloor\dfrac{49+7}{5}\right\rfloor = 11$
$\left\lfloor\dfrac{49}{5}\right\rfloor = 9$
$\left\lfloor\dfrac{7}{5}\right\rfloor = 1$
as you see the equation doesn't hold.
Your intuition is right and the argument is wrong. The same argument should follow the following logic
$\df... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3006557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Test three randomly selected blocks of a program There is one error in one of five blocks of a program. To find the error, we test three randomly selected
blocks. Let $$ be the number of errors in these three blocks. Compute $E(X)$ and $Var(X)$.
That's what i've tried:
$p = 0.6$
$n = 1$
$E(X) = n\cdot p = 1\cdot (0.6)... | Since there is one error in the 5 blocks of code, $X \in \{0,1\}$, where $X=0$ if all 3 blocks of code being tested are correct.
Assuming the error is equally likely to be in any of the 5 blocks of code, the probability of drawing three correct blocks of code is $\frac{4}{5} \times \frac{3}{4} \times \frac{2}{3} = \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3006679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to show $\frac{\Gamma((n-1)/2)}{\Gamma(n/2)} \approx \frac{\sqrt{2}}{\sqrt{n-2}}$
Show $\frac{\Gamma((n-1)/2)}{\Gamma(n/2)} \approx \frac{\sqrt{2}}{\sqrt{n-2}}$
Try
Using the facts:
*
*$(1 + \alpha/m)^m = e^\alpha ( 1+ r_m)$, where $\lim_{m \to \infty} \sqrt{m}r_m = 0$
*$\Gamma(n+1) = n^{n + 1/2} e^{-n} \sqrt... | A possible approach:
$$ \frac{\Gamma\left(\tfrac{n-1}{2}\right)}{\Gamma\left(\tfrac{n}{2}\right)}=\tfrac{1}{\sqrt{\pi}}\,B\left(\tfrac{n-1}{2},\tfrac{1}{2}\right)=\frac{1}{\sqrt{\pi}}\int_{0}^{1}x^{\frac{n-3}{2}}(1-x)^{-1/2}\,dx=\frac{2}{\sqrt{\pi}}\int_{0}^{1}\frac{x^{n-2}}{\sqrt{1-x^2}}\,dx $$
gives
$$ \frac{\Gamma\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3007166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Find $\alpha$,$\beta$ if $\lim_{x→∞}[ \sqrt {ax^2+2bx+c} - \alpha x -\beta] = 0$ Here is my approach.
Consider;
$$ax^2 + 2bx + c = 0$$
or;
$$ x_{±} = \frac {-b±\sqrt {b^2-ac}}{a} = \frac {-b±\sqrt D}{a}$$
Hence;
$$\sqrt {ax^2+2bx+c} = \sqrt {(x+\frac {b-√D}{a})(x+\frac {b+√D}{a})}$$
$$=x\sqrt {(1+\frac {b-√D}{ax})(1+\f... | Set $1/x=h$ to find $$\lim_{h\to0^+}\dfrac{\sqrt{a+bh+ch^2}-(\alpha+\beta h)}h$$
$$=\lim_{h\to0^+}\dfrac{(a+bh+ch^2)-(\alpha+\beta h)^2}h\cdot\lim_{h\to0^+}\dfrac1{\sqrt{a+bh+ch^2}-(\alpha+\beta h)}$$
$(a+bh+ch^2)-(\alpha+\beta h)^2=a-\alpha^2+h(b-2\alpha\beta)+h^2(c-\beta^2)$
As the denominator $\to0,$ $$a-\alpha^2=0$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3007306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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All real number for which $n$ in $5^n+7^n+11^n=6^n+8^n+9^n$
Finding all real number $n$ in
$$5^n+7^n+11^n=6^n+8^n+9^n$$
Try: From given equation
$n=0,1$ are the solution
But i did not understand any other solution exists or not
Although i have tried like this way
$$\bigg(\frac{5}{9}\bigg)^n+\bigg(\frac{7}{9}\bigg)^n... | Consider the function $f(x)=x^n$ for positive $x$. Its second derivative is $n(n-1)x^{n-2}$ and therefore for $n > 1$ or $n<0$ $f$ is strictly convex while for $0 < n < 1$ f is strictly concave.
Our equation is equivalent to $f(5) + f(7) + f(11) = f(6) + f(8) + f(9)$
In the convex case we have:
$f(6) = f(\frac{5+7}{2})... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3007428",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Calculate the following limit without L'Hopital $\lim\limits_{x\to 0}\frac{e^x-\sqrt{1+2x+2x^2}}{x+\tan (x)-\sin (2x)}$ I know how to count this limit with the help of l'Hopital rule. But it is very awful, because I need 3 times derivate it. So, there is very difficult calculations. I have the answer $\frac{2}{5}$.
I ... | $$\begin{eqnarray*} \frac{e^x-\sqrt{1+2x+2x^2}}{x+\tan (x)-\sin (2x)}
& = & \underbrace{\frac{1}{e^x+\sqrt{1+2x+2x^2}}}_{\mbox{harmless}}\cdot \frac{e^{2x}-(1+2x+2x^2)}{x+\tan (x)-\sin (2x)} \\
& \stackrel{\mbox{Taylor}}{=} & \frac{1}{e^x+\sqrt{1+2x+2x^2}}\cdot \frac{\frac{4}{3}x^3+o(x^4)}{\frac{5}{3}x^3+o(x^4)} \\
&\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3007785",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
An equilateral triangle is inscribed in a circle of radius $r$. If $P$ is any point on the circumference, find the value of $PA^2 + PB^2 + PC^2$.
An equilateral triangle is inscribed in a circle of radius $r$. If $P$ is any point on the circumference find the value of $PA^2 + PB^2 + PC^2$.
I have managed to solve ... | Without the loss of generality, we can assume that $P$ is between $A$ and $B$. Thus $m\angle{AOP}+m\angle{BOP}=120°$. Next, either $m\angle{AOP} \ge m\angle{BOP}$ or $m\angle{AOP} < m\angle{BOP}$. Let's assume $m\angle{AOP} \ge m\angle{BOP}$ (this we can do due to the symmetry of the following proof). Then $m\angle{COP... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3008129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 3
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Contour intergal of a rational trigonometric function, can't find my mistake. This is the integral :
$$I = \int_{0}^{2\pi} \frac{dx}{(5+4\cos x)^2}\ $$
Which, according to wolfram alpha, should evaluate to $\frac{10\pi}{27} $,
but the value i find is $\frac{20\pi}{27} $.
These are my calculations :
Using complex form ... | Your mistake is when using the residue theorem:
$$\oint_{C^+} f(z) dz = 2\pi i\sum_{singularities} Res(f,singularity)$$
You forgot the factor $2$ in this formula.
Another mistake: $$5t + 2t^2 + 2 = 2(t+2)(t+1/2)$$
You again forgot a factor 2 (which will yield a factor $4$ eventually)
You also made a mistake in calcul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3008258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Condition in terms of b and a if $ax^2+bx+c=0$ has two consecutive odd positive integers as roots
The roots of the equation$$ax^2+bx+c=0$$, where $a \geq 0$, are two consecutive odd positive integers, then
(A) $|b|\leq 4a$
(B) $|b|\geq 4a$
(C) $|b|=2a$
(D) None of these
My attempt
Let p and q be the roots then if th... | Using the quadratic formula, we have that (using $p<q$ as the roots):
$$p+2=q$$
$$\frac{-b-\sqrt{b^2-4ac}}{2a}+2=\frac{-b+\sqrt{b^2-4ac}}{2a}$$
$$\frac{-b}{2a}-\frac{\sqrt{b^2-4ac}}{2a}+2=\frac{-b}{2a}+\frac{\sqrt{b^2-4ac}}{2a}$$
$$2=2\frac{\sqrt{b^2-4ac}}{2a}$$
$$2a=\sqrt{b^2-4ac}$$
$$4a^2=b^2-4ac$$
$$b^2=4a^2+4ac$$
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3008382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
How to solve $\int_0^{\infty}\frac{\log^n(x)}{1+x^2}dx$? As an exercise for myself I constructed the Integral
$$
\int_0^{\infty}\frac{\log^n(x)}{1+x^2}dx
$$
with $n\in \mathbb{N}$. With the help of Mathematica I found the analytical result
$$
\int_0^{\infty}\frac{\log^n(x)}{1+x^2}dx=\frac{1+(-1)^n}{4^{n+1}}\Gamma(n+1... | Let, for $n$, a natural integer,
\begin{align}J_n=\int_0^\infty \frac{\ln^n x}{1+x^2}\,dx\end{align}
First, observe that if $n$ is odd then $J_n=0$ (perform the change of variable $y=\dfrac{1}{x}$ )
Consider, for $n$, a natural integer,
\begin{align}K_n=\int_0^\infty \int_0^\infty\frac{\ln^n(xy)}{(1+x^2)(1+y^2)}\,dx\,d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3010205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How do I simplify $\frac{\sin(2x)}{1-\cos (2x)}$? $$\dfrac{\sin(2x)}{1-\cos (2x)}$$
How do I simplify given expression?
My attempt:
We know the double-angle identity for $\sin(2x)$ and $\cos(2x)$ as shown below
$$\sin(2x) = 2\sin (x)\cos(x)$$
$$\cos(2x) = 2\cos(x)-1$$
So we have that
$$\dfrac{2\sin (x)\cos(x)}{1-2\cos... | We have
$$\cos(2x) = \cos^2x-\sin^2x=2\cos^\color{red}2(x)-1=1-2\sin^2 x$$
and then
$$\dfrac{\sin(2x)}{1-\cos (2x)}=\dfrac{2\sin x \cos x}{1-1+2\sin^2 x}=\cot x$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3010796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Recurrence k-th pattern I am trying to solve this recurrence $T(n) = 6 T(\frac{n}{3}) + n$.
1st recurrence: $6^2T(\frac{n}{3^2}) + \frac{6n}{3} + n$
2nd: $6^3T(\frac{n}{3^3}) + \frac{6^3n}{3^2} + n$
3rd: $6^4T(\frac{n}{3^4}) + \frac{6^6n}{3^3} + n$
I am having trouble describing the general pattern after the k-th itera... | I assume you are looking to find $T(n)$ after K iterations. First
$$ T(n) = 6 T(n/3) + n$$ then rewriting $T(n/3)$ in terms of $T(n/3^2)$ we conclude:
$$
T(n) = 6^2 T(n/3^2) + 6n/3 +n
$$
similarly $T(n)$ after K iterations becomes:
$$
T(n) = 6^K T(n/ 3^K) + \sum_{i=1} ^K 6^{i-1}n/3^{i-1}
$$
or
$$
T(n) = 6^K T(n/ 3^K... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3011002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proving $(n+1)^2+(n+2)^2+...+(2n)^2= \frac{n(2n+1)(7n+1)}{6}$ by induction My question: $(n+1)^2+(n+2)^2+(n+3)^2+...+(2n)^2= \frac{n(2n+1)(7n+1)}{6}$
My workings
*
*LHS=$2^2$ =$4$ RHS= $\frac{24}{6} =4 $
*$(k+1)^2+(k+2)^2+(k+3)^2+...+(2k)^2= \frac{k(2k+1)(7k+1)}{6}$
*LHS (subsituting $n= k+1$)----> $(k+2)^2+(k+3)^... | $(k+2)^2+\dots+(2k)^2+(2k+1)^2+(2k+2)^2=\frac{k(2k+1)(7k+1)}6-(k+1)^2+(2k+1)^2+(2k+2)^2=\frac{k(14k^2+9k+1)+6(7k^2+10k+4)}6=\frac{(k+1)(2k+3)(7k+8)}6$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3011450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Solve the Differential equation $\left(y e^{\sin x}\cos x-y^3+2xy\right)dx+\left(2e^{\sin x}-4y^2(x+1)+2x^2\right)dy=0$ Solve the equation
$$\left(y e^{\sin x}\cos x-y^3+2xy\right)dx+\left(2e^{\sin x}-4y^2(x+1)+2x^2\right)dy=0$$
My try:
Letting $e^{\sin x}=t$ we have $e^{\sin x}\cos xdx=dt$
so have modified the equatio... | If you multiply the expression on the left-hand side by $y$, you get the exact differential,
$$ y \times \left(\left(y e^{\sin x}\cos x-y^3+2xy\right)dx+\left(2e^{\sin x}-4y^2(x+1)+2x^2\right)dy \right) =d\left( y^2 e^{\sin x} - y^4 (x + 1) + x^2 y^2\right).$$
So the solution is
$$ y^2 e^{\sin x} - y^4 (x + 1) + x^2 y^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3018352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Find the Limit $\lim_{n \to \infty}\sum_{k=1}^{\infty}\frac{k^{n}}{1+k^{n+2}}$ Find $\lim_{n \to \infty}\sum_{k=1}^{\infty}\frac{k^{n}}{1+k^{n+2}}$
My ideas: let $ n \in \mathbb N$ be constant, looking at $\frac{k^{n}}{1+k^{n+2}}$, we know
$$\frac{k^{n}}{1+k^{n+2}}\leq\frac{k^{n}}{k^{n+2}}=\frac{1}{k^{2}}$$
but this do... | The series equals
$$\frac{1}{2}+ \sum_{k=2}^{\infty}\frac{1}{k^{-n} + k^2}.$$
For each $k\ge 2,$ the terms in the last series increase to $1/k^2$ as $n\to \infty.$ By the monotone convergence theorem, the desired limit is
$$\frac{1}{2} +\sum_{k=2}^{\infty}\frac{1}{k^2} = \frac{1}{2}+\left (\frac{\pi^2}{6} - 1\right) = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3024376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Seeking Methods to solve $\int_{0}^{\frac{\pi}{2}} \ln\left|\sec^2(x) + \tan^4(x) \right|\:dx $ After weeks of going back and forth I've been able to solve the following definite integral:
$$I = \int_{0}^{\frac{\pi}{2}} \ln\left|\sec^2(x) + \tan^4(x) \right|\:dx $$
To solve this I employ Feynman's Trick with Glasser's ... | Here's an alternative solution. First, notice that if $I(a,b) = \int_0^\pi \ln\left(a + b\cos(x)\right) \mathrm{d}x$ then $I(a,0) = \pi \ln(a)$. Thus
\begin{align}
\frac{\partial I}{\partial a}
&=\int_0^\pi \frac{\mathrm{d}x}{a + b\cos(x)} \overset{t=\tan\left(\frac{x}{2}\right)}{=}\frac{2}{a-b}\int_0^\infty \frac{\mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3026362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 2
} |
What is the area shaded in this figure? I was sent this problem by a friend, they say it is a year 6 question. Using trigonometry I got the shaded area as $4 - \frac{\pi}{3}$.
Now Year six, I doubt if they do trigonometry. Is there another approach out there?
|
Label the vertices as shown in the diagram.
$\tan\alpha=\frac{1}{2}\implies \tan 2\alpha=\frac{4}{3}\implies 2\alpha=\tan^{-1}(\frac{4}{3})$
$\therefore$ Area of circular sector$ BAG=8\tan^{-1}(\frac{4}{3})$
Area of $\triangle AGC=16\cos\alpha\sin\alpha=8\sin2\alpha=\frac{32}{5}$
Thus, area of curved figure $BEG=\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3032164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Average power of 2 in all even natural numbers Consider all even natural numbers.
Every 4th number has a power of 4 (or $2^2$)
Every 8th number has a power of 8 (or $2^3$)
Every 16th number has a power of 16 (or $2^4$)
What is the average number of power of 2 in any random even number?
Since every 2nd number has a powe... | A multiple of $4$ only has two powers of $2$ in its factorization and a multiple of $8$ only has three. Your sum should therefore be
$$\left(1 \times \dfrac{1}{2}\right) + \left(2 \times \dfrac{1}{4}\right) + \left(3 \times \dfrac{1}{8}\right) + \ldots=2$$
because half the evens have exactly one factor of two, one q... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3032706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Calculate $\sin\frac{\pi}{16}$ from given trigonometric identities There is the choice between four trigonometric identities
$$\sin(4\phi)=8\cos^{3}(\phi)\sin(\phi)-4\cos(\phi)\sin(\phi)$$
$$\cos(4\phi)=8\cos^{4}(\phi) -8\cos^{2}(\phi)+1$$
$$\sin^{4}(\phi)=1/8(\cos(4\phi) -4\cos(2\phi)+3)$$
$$\cos^{4}(\phi)=1/8(\cos(4\... | Use the second identity:
$$ 8\cos^4\left(\frac{\pi}{16}\right) - 8\cos^2\left(\frac{\pi}{16}\right) + 1 = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt2}{2} $$
Let $x = \cos^2\left(\frac{\pi}{16}\right)$ then
$$ 8x^2-8x = \frac{\sqrt{2}-2}{2} $$
Completing the square:
$$ 4x^2 - 4x + 1 = \frac{\sqrt{2}-2}{4}+1 = \frac{2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3035536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Finding a recurrence relation I need to find a recurrence relation or an exact formula to the sequence
$$1,2,4,8,12,16,24,32,40,48,64,80,96,112,128,160,192,224,256,288,\ldots$$
Well, considering $a_0=1$, $a_1=2$ and $a_2=4$, the terms $a_n$ for $n\geq3$ is obtained by adding a power of $2$. In fact the sequence is:
$$1... | To compute the $n^{th}$ term, find $k$ such that $\frac{k(k-1)}{2} \leq n \leq \frac{k(k+1)}{2}$. Then:
$x(n) = x(n-1) + 2^{k-1}$.
For example, if $n=9$, $k=4$ and $x(9) = x(8) + 8$.
Let's first look at the roots of $k^2-k -2n=0$. They are given by $r_1 = \frac{1-\sqrt{1+8n}}{2}$ and $r_2 = \frac{1+\sqrt{1+8n}}{2}$.
T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3036361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How many pairs $(x,y)$ such that $x+y=n$ have an $x$ or $y$ divisible by 3 but $x$ and $y$ are not equal to 3? Let the set $S_{n}$ = {$(x,y):x,y \in \mathbb{O}$} such that $x+y=n$ where $\mathbb{O}$ is set of odd integers > 1.
Let us define the function $f(n) = |S_{n}|$ that counts the number of pairs in $S_{n}$.
Examp... | The sum of $2$ multiples of $k$ is also a multiple of $k$. Therefore, if $n$ is not divisible by $3$, we know that the $\left\lfloor \frac{n}{6}\right\rfloor -1$ values of $p$ where $p$ is divisible by $3$ will not overlap any of the $\left\lfloor \frac{n}{6}\right\rfloor -1$ values of $q$ where $q$ is divisible by $3$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3038250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that the $L^{p}$ norm $\|f\|_{L^{p}} := \big( \int^{b}_{a} |f(x)|^p\big)^{1/p}$ is not induced by a scalar product for $p \neq 2$.
On $X = C^0\big([a,b]\big)$, for any $p \in \mathbb{R}$, $p>1$, we define the $L^p$ norm by,
$$\|f\|_{L^{p}}:=\big(\int^{b}_{a}|f(x)|^{p}dx \big)^{1/p}.$$
Show that for $p\neq 2$,... | Here is a similar idea with simpler computation. Define
$$f(x) = \begin{cases} \left(\frac{a+b}2-x\right)^{1/p},&\text{ if } x \in \left[a, \frac{a+b}2\right] \\
0,&\text{ if } x \in \left[\frac{a+b}2,b\right]
\end{cases}$$
$$g(x) = \begin{cases} 0,&\text{ if } x \in \left[a, \frac{a+b}2\right] \\
\left(x-\frac{a+b}2\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3039866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Compute the limit $\lim_{n\to\infty} I_n(a)$ where $ I_n(a) :=\int_0^a \frac{x^n}{x^n+1}\,\mathrm{d}x, n\in N$. For $a>0$ we define
$$\space I_n(a)=\int_0^a\frac{x^n}{x^n+1}\,\mathrm{d}x , n\in N.$$
*
*Prove that $0\le I_n(1) \le \frac{1}{n+1}$
*Compute $\lim_{n\to\infty} I_n(a)$
My attempt:
*
*I regard $I_n(1)... | Certainly not the most compact approach, but:
\begin{equation}
I_n(a) = \int_{0}^{a} \frac{w^n}{w^n + 1}\:dw = \int_{0}^{a}\left[ 1 - \frac{1}{w^n + 1}\right]\:dw = a - \int_{0}^{a}\frac{1}{w^n + 1}\:dw
\end{equation}
Now:
\begin{equation}
J_n(a) = \int_{0}^{a} \frac{1}{w^n + 1}\:dw
\end{equation}
With $n \geq 1$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3042601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Proof verification: For $a$, $b$, $c$ positive with $abc=1$, show $\sum_{\text{cyc}}\frac{1}{a^3(b+c)}\geq \frac32$ I would like to have my solution to IMO 95 A2 checked. All solutions I've found either used Cauchy-Schwarz, Chebyshevs inequality, the rearrangement inequality or Muirheads inequality. Me myself, I've use... | Your proof is nearly correct: $$f''(x)=\frac{2S^2}{(S-x)^3}$$ (the exponent is $3$, not $2$). This is still okay because $x,y,z>0$, but you need a bit more justification.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3043074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"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.