All Questions
274
questions with no upvoted or accepted answers
7
votes
0
answers
110
views
Summation $\sum_{n=0}^{\infty}\left(\frac{x^{2n}\left(n+1\right)^{x}\ }{n!}\right)$
$$f\left(x\right)=\sum_{n=0}^{\infty}\left(\frac{x^{2n}\left(n+1\right)^{x}\ }{n!}\right)$$
This is a self made problem I came across while researching something, so I do not have high hopes for it ...
- 3,280
7
votes
1
answer
215
views
Help Calculation of $\sum_{k=1}^{\infty} \frac{k^{2n}}{e^k -1}$
Recently, I read a book : Euler, Riemann, Ramanujan - Contact mathematician beyond the space-time by Nobushige Kurokaw. It says that Ramanujan had found the following formula
$$\sum_{k=1}^{\infty} \...
- 161
7
votes
0
answers
453
views
Can we interchange the Integral and Summation when a limit is $\infty$?
I was trying to Evaluate the Integral:
$$\Large{I=\int_1^{\infty} \frac{\ln x}{x^2+1} dx}$$
$$\color{#66f}{{\frac{1}{x^2+1} = \frac{1}{x^2\left(1+\frac{1}{x^2}\right)}=\frac{1}{x^2}\cdot \frac{1}{1+...
- 3,064
6
votes
0
answers
535
views
Integral $I=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx$
Hi I am trying to integrate and obtain a closed form result for
$$
I:=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx.
$$
Here is what I tried (but I do not think this is ...
- 9,976
5
votes
0
answers
134
views
Evaluate the sum : $\frac{1}{1+x_{1}}+\frac{1}{1+x_{2}}+\frac{1}{1+x_{3}}+...+\frac{1}{1+x_{n}}$
Question :
Let the real number $x≥1$ :
$x_{1}=x$ and $x_{n+1}=x_{n}(1+x_{n})$ for
$n=1,2,3...$
Then find the sum :
$S=\displaystyle \sum_{k=1}^{n}\frac{1}{1+x_{k}}$
My try :
Note that : $...
- 2,323
5
votes
1
answer
78
views
Find the minimum posssible integer value of the summation
Let $f(x)$ is a continuous, increasing and positive value function in the interval $[0,a]$ such that
$$\int_0^af(x)dx=20$$
Then find the minimum posssible integer value of the following summation
$$a\...
- 506
4
votes
0
answers
135
views
Simplify a summation in the solution of $\displaystyle\int_{0}^{\infty}e^{-cx}x^{n}\arctan(ax)\mathrm{d}x$
Context
I calculated this integral:
$$\begin{array}{l}
\displaystyle\int_{0}^{\infty}e^{-cx}x^{n}\arctan(ax)\mathrm{d}x=\\
\displaystyle\frac{n!}{c^{n+1}}\left\lbrace\sum_{k=0}^{n}\left[\text{Ci}\left(...
4
votes
0
answers
101
views
How to compute $-\sum_{n=0}^{\infty}{\frac{(2n)!}{2^{2n}((n+1)!)^{2}}}$
I'm trying to compute $\int_{0}^{1}{\frac{\ln(x)}{\sqrt{1-x}}}dx$.
I already computed this integral using substitution and IBP and it's value is $4(\ln(2)-1)$.
When I use the serie expansion of $\frac{...
- 41
4
votes
0
answers
142
views
Rare sum with Trigamma function $\left( -1 \right)^{n-1}\left( \psi _{1}\left( n \right) \right)^{2}$
$$\sum\limits_{n=1}^{+\infty }{\left( -1 \right)^{n-1}\left( \psi _{1}\left( n \right) \right)^{2}}$$
Here is my procedure, but I do not know how to calculate the last integral. Anyone have any ideas?...
- 765
4
votes
0
answers
289
views
Extending Sophomore's Dream
I've been trying to extend the famous Sophomore's Dream identity by allowing the upper limit in the integral to be any natural number $k$:
$$\int_0^k x^{-x} dx =\ ? $$
Following the analytical proof ...
- 7,532
4
votes
0
answers
105
views
convenient and precise calculation of $\sum_{n=1}^{\infty}\frac{1}{n^b(n+m)^b}$
I would like to sum the following $$\sum_{n=1}^{\infty}\frac{1}{n^b(n+m)^b}$$
for values of $b$ for which the sum exists. The problem is that truncating the sum does not work that good, and needs in ...
- 9,677
4
votes
0
answers
55
views
Questionable Convergence of a Series
The summation is:
$$
S = \sum_{k \geq 0} f(k) \int_{0}^{\pi/2} \sqrt{1-(1- \frac{f(k+1)^2}{f(k)^2})\sin^2(\theta)}d\theta
$$
Now, we know that $f(k+1) < f(k)$ and as $k$ approaches infinity, $f(...
- 1,957
3
votes
0
answers
139
views
Evaluate the sum $S = \sum\limits_{n = 0}^\infty {{{\left( { - 1} \right)}^n}{\text{Ci}}\left( {\pi \left( {2n + 1} \right)} \right)} $
I am trying to evaluate this sum:$$S = \sum\limits_{n = 0}^\infty {{{\left( { - 1} \right)}^n}{\text{Ci}}\left( {\pi \left( {2n + 1} \right)} \right)} $$
$${\text{Where}}:{\text{Ci}}\left( x \right){\...
- 2,702
3
votes
0
answers
107
views
Series representation of $n$th derivative of $x^n/(1+x^2)$
Find the nth derivative of $\frac{x^n}{1+x^2}$.
Please I need help in this. They are further asking to show that when $x=\cot y$ the nth derivative can be expressed as
$$n!\sin y\sum_{r=0}^{n}(-1)^r {...
3
votes
0
answers
107
views
Proving double sum $\sum _{a=1}^{\infty }\sum _{b=1}^{\infty }\frac{\left(-1\right)^{a+b}}{\left(a^2+b^2\right)^s} $
I have see this formula in Wolframe Alpha Double series's documentation, but what is the procces of proving?
$$\sum _{a=1}^{\infty }\sum _{b=1}^{\infty }\frac{\left(-1\right)^{a+b}}{\left(a^2+b^2\...
- 142