All Questions
86
questions
6
votes
4
answers
500
views
Finding $\sum_{n=1}^{\infty}\frac{(-1)^n (H_{2n}-H_{n})}{n2^n \binom{2n}{n}}$
I want to find the closed form of:
$\displaystyle \tag*{} \sum \limits _{n=1}^{\infty}\frac{(-1)^n (H_{2n}-H_{n})}{n2^n \binom{2n}{n}}$
Where $H_{k}$ is $k^{\text{th}}$ harmonic number
I tried to ...
- 921
1
vote
0
answers
74
views
Finding the value of $ \sum_{n=1}^{\infty} \frac{2(2n+1)}{\exp( \frac{\pi(2n+1)}{2})+\exp ( \frac{3\pi(2n+1)}{2})}$
I have a question which askes to find the value of:
$$\displaystyle \tag*{} \sum \limits_{n=1}^{\infty} \dfrac{2(2n+1)}{\exp\left( \dfrac{\pi(2n+1)}{2}\right)+\exp \left( \dfrac{3\pi(2n+1)}{2}\right)}...
- 921
1
vote
1
answer
115
views
Why is this sum equal to an integral?
I participate the stochastic course and we now speak about summable families. There we have the following definition:
Let $\Omega$ be countable and $a:\Omega\rightarrow \Bbb{R}_+\cup \{\infty\}$ be a ...
- 2,935
0
votes
1
answer
84
views
Inequality involving finite sum and integral
I'm reading a proof where they use the following inequality:
$$\sum_{k=4}^n \frac{k^2}{n}(1-a)^{k+1}\le\int_0^\infty \frac{x^2}{n}\exp{(-ax)}$$
For $a>0$. I'm trying to show it.
So far I got
$$\...
- 376
1
vote
1
answer
56
views
Calculating bounds for a certain expression using Faulhaber
Let $F(n,k)=\sum_{i=1}^ni^k$ for $k\geq 0$ and $n\geq 1$. I would like to prove
$$\sum_{i=1}^{n-1}i^k\leq\frac{n^{k+1}}{k+1}\leq\sum_{i=1}^ni^k.$$
The idea is to use integration where the upper and ...
- 43
9
votes
6
answers
376
views
Evaluate : $S=\frac{1}{1\cdot2\cdot3}+\frac{1}{5\cdot6\cdot7}+\frac{1}{9\cdot10\cdot11}+\cdots$
Evaluate:$$S=\frac{1}{1\cdot2\cdot3}+\frac{1}{5\cdot6\cdot7} + \frac{1}{9\cdot10\cdot11}+\cdots$$to infinite terms
My Attempt:
The given series$$S=\sum_{i=0}^\infty \frac{1}{(4i+1)(4i+2)(4i+3)} =\sum_{...
- 9,599
6
votes
1
answer
260
views
Formula for $f(1) + f(2) + \cdots + f(n)$: Euler-Maclaurin summation formula
Let $f\colon \mathbb{R}\to \mathbb{R}$ be a function with $k$ continuous derivatives. We want to find an expression for
$$
S=f(1)+f(2)+f(3)+\ldots+f(n).
$$
I'm currently reading Analysis by Its ...
- 435
2
votes
1
answer
230
views
Unable to solve integral of a summation
I want to integrate each datapoint in a 2D image $X$ over a disc, but i'm really unsure what to do with the summation term. Essentially, at each point $x_j$, we integrate a disc of radius $r$, ...
- 73
1
vote
2
answers
201
views
Determine if $\int_1^\infty \arctan(e^{-x})dx$ converges or diverges
I want to check if the sequence $\sum_{n=1}^\infty [\sin(n) \cdot \arctan(e^{-n})]$ converges absolutely, by condition, or not.
We know that $\sum_{n=1}^\infty [\sin(n) \cdot \arctan(e^{-n})] \le \...
- 627
2
votes
1
answer
45
views
Replacement of the tagged partition displacement of the Riemann Integral
In specific, my question is if I can make the following by the definition on Riemann integral...
$$\int_a^bf(x)dx = \lim_{\lambda \rightarrow 0}\sum_{i=1}^n f(c_i)\Delta x_i = \lim_{\lambda \...
- 59
1
vote
0
answers
48
views
Value of a summation is greater than the corresponding integration
Let $$x[n] = 2^{-n}u[n + 1]$$ $$h[n] = 3^{n}u[-n+2]$$If we perform convolution, we get $$y[m]=x[n]*h[n] = \sum_{l=-\infty}^{+\infty}x[l]h[m-l] = \cases{\frac{27\times2^{3 - m}}{5} \ m \ge 2 \\ \frac{...
- 4,359
0
votes
1
answer
73
views
Integration and summation inequality
This step arise while proving infinite Hilbert matrix is bounded linear operator using Schurz test.
In my book (Functional Analysis by S Kesavan) it's given as follows:
$$ \sum_{i=0}^\infty \frac{1}{(...
- 2,506
1
vote
2
answers
100
views
Finding a closed form expression of a sequence that is defined recursively via a definite integral
Consider the following series function that is defined recursively by the following definite integral
$$
f_n(x) = \int_0^x u^n f_{n-1}(u) \, \mathrm{d}u \qquad\qquad (n \ge 1) \, ,
$$
with $f_0 (x) = ...
0
votes
1
answer
216
views
Is it possible to interchange sum and integral when the series doesn't converge uniformly?
Suppose a series $\sum f_n$ of integrable functions on $[0,1]$ does not converge uniformly, is it possible that $$\sum_{n=1}^\infty \int_0^1 f_n(x)dx = \int_0^1\sum_{n=1}^\infty f_n(x)dx$$ still holds?...
- 225
1
vote
3
answers
71
views
Bounding sum by (improper) integral
I am trying to verify the following inequality that I came across while reviewing some analysis exercises online:
$$
\sum_{n=1}^{k} \left(1-\frac{n}{k}\right)n^{-1/7}\leq \int_{0}^{k}\left(1-\frac{x}{...
- 35