All Questions
11
questions
40
votes
7
answers
5k
views
How to evaluate $\sum_{k=3}^{\infty} \frac{\ln (k)}{k^2 - 4}$?
Is it possible to evaluate the sum:
$$\sum_{k=3}^{\infty} \frac{\ln (k)}{k^2 - 4}$$
I expect it may be related to $\zeta^{\prime} (2)$:
$$\zeta^{\prime} (2) = - \sum_{k=2}^{\infty} \frac{\ln(k)}{k^2}$$...
- 5,337
4
votes
4
answers
171
views
How to prove that $ \int_{0}^{\infty} x^n e^{-xt} \sin x\frac{dx}{x} =\frac{i(n-1)!}{2(1+t^2)^n}\left((t-i)^n-(t+i)^n\right)$
Let $n\ge 1$ How to prove that $$I_n(t)= \int_{0}^{\infty} x^n e^{-xt} \sin x\frac{dx}{x} =\frac{(n-1)!}{(1+t^2)^n}\frac{\left((t+i)^n-(t-i)^n\right)}{2i}$$
I have manage to prove that one can ...
- 24.2k
4
votes
2
answers
95
views
Is there a close form of $ S_n =\sum_{k=0}^{\infty}\frac{k^n}{k!}$
I want to fine the close form of the following sequence
$$S_n =\sum_{k=0}^{\infty}\frac{k^n}{k!}$$
here is my attempt ,
$$S_0 = \sum_{k=0}^{\infty}\frac{1}{k!} =e~~~\text{and }~~~S_1 = \sum_{k=1}^{...
- 24.2k
7
votes
3
answers
200
views
Is there any closed form for this integral?
The picture above is from uninstallation tool of fake antivirus in Korea. The "official" uninstallation tool will not proceed anymore unless user input the correct answer. (Nobody succeeded this) Due ...
- 777
6
votes
2
answers
539
views
Closed form for a series of functions
Let $f_1:\mathbb{R}\to \mathbb{R}$ be a locally integrable function (that is $f_1\in L^1_{loc}(\mathbb{R})$). Let us define $f_{n+1}:=\int_0^x f_n(t)\,dt$ for all $n\ge1$. We consider the series $S(x)=...
- 1,283
0
votes
2
answers
229
views
How to show $e^{-\sqrt{2\lambda }x}=\int_0^{\infty} e^{-\lambda u} (2\pi u^3)^{-1/2} x \exp(-\frac{x^2}{2u}) du$? [closed]
I want to show the following integral:
$$e^{-\sqrt{2\lambda }x}=\int_0^{\infty} e^{-\lambda u} (2\pi u^3)^{-1/2} x \exp(-\frac{x^2}{2u}) du$$
Could anyone give me some hints on evaluating the right ...
- 539
8
votes
0
answers
415
views
Improper Integral $\int_0^\infty\tan\left(\frac x{\sqrt{x^3+x^2}}\right)\frac{\ln(1+\sqrt x)}xdx$
This integral is from integral
Find
$$\int_0^\infty\tan\left(\frac x{\sqrt{x^3+x^2}}\right)\frac{\ln(1+\sqrt x)}xdx$$
I have get
$$\int_0^\infty\tan\left(\frac x{\sqrt{x^3+x^2}}\right)\frac{\ln(1+\...
- 93.6k
1
vote
1
answer
36
views
How to show $\sum\limits_{i=1}^{t}\frac{1}{i}2^{t-i}=2^t\ln 2 -\frac{1}{2}\sum\limits_{k=0}^\infty \frac{1}{2^k(k+t+1)}$
How to show the below equation ?
$$\sum\limits_{i=1}^{t}\frac{1}{i}2^{t-i}=2^t\ln 2 -\frac{1}{2}\sum\limits_{k=0}^\infty \frac{1}{2^k(k+t+1)}
~~~~~(t\in \mathbb Z^+)$$
- 1,543
10
votes
1
answer
2k
views
How to calculate $\int_0^\pi \ln(1+\sin x)\mathrm dx$
How to calculate this integral
$$\int_0^\pi \ln(1+\sin x)\mathrm dx$$
I didn't find this question in the previous questions. With the help of Wolframalpha I got an answer $-\pi \ln 2+4\mathbf{G}$, ...
- 5,281
30
votes
3
answers
1k
views
Conjectured closed form for $\sum_{n=-\infty}^\infty\frac{1}{\cosh\pi n+\frac{1}{\sqrt{2}}}$
I was trying to find closed form generalizations of the following well known hyperbolic secant sum
$$
\sum_{n=-\infty}^\infty\frac{1}{\cosh\pi n}=\frac{\left\{\Gamma\left(\frac{1}{4}\right)\right\}^2}{...
user294724
1
vote
3
answers
116
views
Find the limit of the sequence $\left( \sqrt {2n^{2}+n}-\sqrt {2n^{2}+2n}\right) _{n\in N}$
My answer is as follows, but I'm not sure with this:
$\lim _{n\rightarrow \infty }\dfrac {\sqrt {2n^{2}+n}}{\sqrt {2n^{2}+2n}}=\lim _{n\rightarrow \infty }\left( \dfrac {2n^{2}+n}{2n^{2}+2n}\right) ^{\...
- 39