All Questions
249
questions
0
votes
0
answers
46
views
Why can we approximate a sum by a definite integral?
From wikipedia https://en.wikipedia.org/wiki/Summation#Approximation_by_definite_integrals, I read that
$\int_{s=a-1}^{b} f(s)\ ds \le \sum_{i=a}^{b} f(i) \le \int_{s=a}^{b+1} f(s)\ ds$ for increasing ...
- 365
4
votes
2
answers
86
views
Symmetrical sums
$$\sum_{j=1}^{\infty }\left ( \sum_{n=1}^{\infty } \frac{e^{-jx/n}}{n^2}-\frac{e^{-nx/j}}{j^2}\right ),x>0$$
$$\sum_{j=1}^{\infty }\left ( \sum_{n=1}^{\infty }\frac{1}{\sqrt{nj}} \left (\frac{e^{-j/...
- 1,431
1
vote
2
answers
66
views
Closed form for $\sum_{r=0}^{n}(-1)^r {3n+1 \choose 3r+1} \cos ^{3n-3r}(x)\sin^{3r+1} (x)$
I request a closed form for the following sum $$S(n)=\sum_{r=0}^{n}(-1)^r {3n+1 \choose 3r+1} \cos ^{3n-3r}(x)\sin^{3r+1} (x) $$
I tried using De Moivre's theorem $$\cos(nx)+i\sin(nx)=(\cos (x)+i\sin(...
user1103841
0
votes
0
answers
45
views
Evaluate $\sum_{n=1}^\infty \dfrac{\sin n}n$ [duplicate]
Evaluate $\sum_{n=1}^\infty \dfrac{\sin n}n$.
By Euler's theorem, $\sin x = \dfrac{e^{ix}-e^{-ix}}{2i}$ for every real number x. Also, I know that $\sum_{n=1}^\infty \dfrac{1}{n^2} = \dfrac{\pi^2}6.$ ...
- 1,837
3
votes
1
answer
75
views
Asymptotic formula of $\sum_{r=1}^{n}\frac{(-1)^{r-1}}{(r-1)!}{n\choose r}$
I need to find the asymptotic equivalence of the sum $$\sum_{r=1}^{n}\frac{(-1)^{r-1}}{(r-1)!}
{n\choose r} $$ where ${n\choose r}$ is the binomial coefficient.
We have the binomial identity $$(1-x)^...
user1094088
2
votes
1
answer
57
views
prove that the maximum value of a function is attained when all $x_i$'s are 0 or 1
Let $0\leq x_i\leq 1$ for $1\leq i\leq n.$ Prove that the maximum value of the sum $S(x_1,\cdots, x_n) = \sum_{i=1}^n x_i - \sum_{i=1}^n x_i x_{i+1},$ where $x_{n+1} := x_1$ is attained when all $x_i$'...
- 1,137
2
votes
1
answer
47
views
prove the maximum of $\sum_{1\leq r<s\leq 2n}(s-r-n)x_rx_s$ is attained when all $x_i$'s are $-1$ or $1$
Let n be a positive integer. Prove that the maximum possible value of $Z =\sum_{1\leq r<s\leq 2n}(s-r-n)x_rx_s$, where $-1\leq x_i\leq 1$ for all i, is attained when $x_i\in \{-1,1\}\,\forall i$.
...
- 1,837
-1
votes
3
answers
170
views
On an infinite product $\prod_{k=1}^\infty\left(1-\frac{x^3}{k^3\pi^3}\right)$ [closed]
If $$f(x)=\prod_{k=1}^\infty\left(1-\frac{x^3}{k^3\pi^3}\right)$$
and if $$f(x)=\sum_{n=0}^\infty a_n x^n$$
Then find $a_n$ or convert the above infinite product into an infinite sum.
We see that $f(...
user1089939
0
votes
0
answers
56
views
prove that $\sum_{i,j=1}^n f(a_i - a_j) = \int_{-\infty}^\infty (\sum_{i=1}^n \frac{1}{1+(x-a_i)^2})^2 dx,$
Prove that if $a_1,a_2,\cdots, a_n$ are real numbers then $$\sum_{i,j=1}^n f(a_i - a_j) = \int_{-\infty}^\infty \left(\sum_{i=1}^n \frac{1}{1+(x-a_i)^2}\right)^2 dx,$$ where $f(y) = \int_{-\infty}^\...
- 1,137
3
votes
3
answers
135
views
Upper bound of the sum $\sum_{n=1}^\infty \tan^{-1}\left(\frac{x}{1-(x-n)^2}\right)$ as $x\to \infty$
I am stuck at finding a Big-O upper bound of the following sum as $x\to \infty$ $$\sum_{n=1}^\infty \tan^{-1}\left(\frac{x}{1-(x-n)^2}\right)$$
I tried: Since $$ \sum_{n=1}^\infty\tan^{-1}\left(\frac{...
user1082747
0
votes
0
answers
105
views
Manipulation with the following infinite sum
Calculating some observable, I obtained the following-like converges sum
$$
S = \sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\sum_{p=0}^{\min(n,k)} \frac{x^n}{n!} \frac{y^k}{k!} F(p),
$$
where $F$ - some ...
- 121
0
votes
2
answers
285
views
Radius of convergence of $\sum_{n=1}^{\infty}\frac{x^n}{1-x^n}$? [closed]
I'm struggling to find the radius of converge of the following sum:
\begin{equation}
\sum_{n=1}^{\infty}\frac{x^n}{1-x^n}
\end{equation}
When I apply the Dalambre's criterion I get:
\begin{equation}
\...
1
vote
0
answers
20
views
proofs about a norm involving an integral and fourier coefficients
Let $C([-\pi, \pi])'$ be the set of continuous functions $f$ from $[-\pi, \pi]$ to $\mathbb{C}$ such that $f(\pi) = f(-\pi)$. For $f \in C([-\pi, \pi])', n \in \mathbb{Z}$, define $a_n = \frac{1}{2\pi}...
- 1,225
1
vote
0
answers
154
views
prove that there is a continuous and nondecreasing function from $[0,1]$ to itself whose graph has length at least 2
Prove that there exists a function $f : [0,1]\to [0,1]$ that is continuous and nondecreasing so that the length of the graph of $f$ is at least $2$. The length of the graph of $f$ is the supremum, ...
- 1,225
1
vote
1
answer
99
views
Upper bound on the integral of a step function inequality
Problem
I have been working through content in real analysis and came across across an inequality in a proof that is unclear to me.
We are told that a function $\psi$ is a step function on the ...
- 4,331