All Questions
Tagged with integration sequences-and-series
1,992
questions
-5
votes
0
answers
54
views
A simple yet complex proof that I am unable to solve. [closed]
prove that: ((e^(ax ))cos(bx))^n=((sqrt(a^2+b^2))^n)(e^(ax))cos(bx + n.arctan(b/a))
0
votes
0
answers
69
views
Landau Notation Problem
I have this function
$$ K_{n} = \int_{1}^{+\infty}\frac{1}{(1+t^2)^n}dt$$ $$ \text{Let }t\geq1,t^2+1\geq1+t\Leftrightarrow\frac{1}{1+t^2}\leq\frac{1}{1+t} \text{ and for } n \in {\mathbb{N^{*}}} : \...
-1
votes
0
answers
71
views
Need Help Simplifying a Series Involving Exponential and Factorial Terms
I'm working on solving a fractional differential equation and encountered the following series:
$$
\sum_{n = 1}^{\infty}
\frac{\displaystyle\beta^{n}n^{p}\,{\rm e}^{...
1
vote
4
answers
164
views
Calculating: $\lim_{n\to\infty}{\int_{3}^{4}}( \sqrt{-x^2+6x-8})^{n}dx$ [duplicate]
Calculate: $\lim_{n\to\infty}\int_{3}^{4} \left(\sqrt{-x^2+6x-8}\right)^{n}{\rm d}x.$
I've tried to change the variable and I took it as $y=x-3$.
I have changed the limits of integrations, for $x=3$, ...
- 163
3
votes
3
answers
386
views
$\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$ as a limit of a sum
Working on the same lines as
This/This and
This
I got the following expression for the Dilogarithm $\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$:
$$\operatorname{Li}_{2} \left(\frac{1}{e^{\...
- 862
2
votes
1
answer
88
views
If we know the even fourier series are we able to find the odd version?
If we have a sequence, call it $a_{n}$ where $n>0$, and we take it as the fourier coeficents of an even function...
$$
F_{e}(x) = \sum^{\infty}_{n=1}a_{n}\cos(2\pi nx)
$$
... and we know the form ...
- 718
-1
votes
1
answer
61
views
A problem involving series and integration [duplicate]
Suppose that $f:[0,\infty)\rightarrow \mathbb C$ is a $C^1$ function satisfying $f(0)=0$ and $\int_0 ^\infty (|f(y)| + |f^{'}(y)|)dy<\infty$. Show that
$$\left\vert \sum_{0\leq n< \infty} f(n) - ...
- 750
0
votes
0
answers
96
views
$\operatorname{Li}_{2} \left(\frac12 \right)$ vs $\operatorname{Li}_{2} \left(-\frac12 \right)$ : some long summation expressions
Throughout this post, $\operatorname{Li}_{2}(x)$ refers to Dilogarithm.
While playing with some Fourier Transforms, I came up with the following expressions:
$$2 \operatorname{Li}_{2}\left(\frac12 \...
- 862
14
votes
4
answers
586
views
Closed form: show that $\sum_{n=1}^\infty\frac{a_n}{(n+1)4^n}=\zeta(2) $
Let $a_n$ be the sequence defined via
$$ a_n=\sum_{k=1}^n{2n \choose {n-k}}\frac{1+(-1)^{k+1}}{k^2}$$
then prove that
$$\sum_{n=1}^\infty\frac{a_n}{(n+1)4^n}=\frac{\pi^2}{6} $$
I tried simplifying $...
- 3,402
1
vote
1
answer
105
views
Evaluate the infinite summation and find $L$
Evaluate
$$
L=\sum_{r=1}^{\infty}\frac{1}{2^r\left(1 + 2^{1/2^{\LARGE r}}\right)}
$$
At first glance, I thought the limit could be solved by converting it to a ...
3
votes
1
answer
67
views
Compute the integral: $\int_{0}^{\infty}e^{-x}J_{0}(x)dx$ [duplicate]
At first, I was computing the following integral: $$\int_{0}^{\infty}e^{-x}J_0(\sqrt{x})dx$$ which can be easily solved using the taylor series of $J_0(x)$:
$$J_0(x)=\sum_{n=0}^{\infty}\frac{(-1)^nx^{...
- 861
3
votes
1
answer
94
views
Calculate $ \lim_{n \to \infty} \left( \sum_{k=1}^n \left( \sqrt{n^4 + k} \cdot \sin \left( 2\pi \cdot \frac{k}{n} \right) \right) \right) \ $
$$
\mbox{What is the value of this limit ?:}\quad
\lim_{n \to \infty}\sum_{k = 1}^{n}\sqrt{n^{4} + k\,}\
\sin\left(2\pi\,\frac{k}{n}\right)
$$
I tried looking for sum Riemann sums first, nothing.
I ...
- 1,311
4
votes
1
answer
91
views
Why $\infty=\sum_{i=1}^\infty \frac{1}{n+i}\neq\lim_{n\rightarrow\infty}\sum_{i=1}^n \frac{1}{n+i}=\log2$?
I was wondering why $\sum_{i=1}^\infty \frac{1}{n+i}$ diverges but $\lim_{n\rightarrow\infty}\sum_{i=1}^n \frac{1}{n+i}=\log2$. While assuming integral as limit of series, we find out that:
$$
\int_1^...
- 43
2
votes
1
answer
113
views
Calculate $\sum\limits_{n=0}^{\infty}{\int\limits_{\frac{1}{2}}^{\infty}(1-e^{-t})^{n}e^{-t^2}dt}$
$$
\mbox{Calculate}\quad
\sum_{n = 0}^{\infty}\int_{1/2}^{\infty}
\left(1 - {\rm e}^{-t}\right)^{n}{\rm e}^{-t^{2}}{\rm d}t
$$
Basically I don't know where to start.
I was thinking of using ...
- 187
1
vote
1
answer
38
views
Compute $\int_{\phi_1}^{\phi_2} \left( \sum_{n=0}^N a_n \cos(n \phi) \right)^2 \,\text{d}\phi$ analytically?
For a computational model I need to evaluate the following integral many times:
$$
I(\phi_1, \phi_2) = \int_{\phi_1}^{\phi_2} \left( \sum_{n=0}^N a_n \cos(n \phi) \right)^2 \,\text{d}\phi,
$$
where I ...
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