All Questions
Tagged with polylogarithm power-series
9
questions
12
votes
2
answers
499
views
How to determine the value of $\displaystyle f(x) = \sum_{n=1}^\infty\frac{\sqrt n}{n!}x^n$?
How to determine the value of $\displaystyle f(x) = \sum_{n=1}^\infty\frac{\sqrt n}{n!}x^n$? No context, this is just a curiosity o'mine.
Yes, I am aware there is no reason to believe a random power ...
- 3,871
4
votes
1
answer
82
views
Evaluate the following series sum.
Problem
I’m trying to evaluate the following series sum
\begin{equation}
S_{j}(z) = \sum_{k=1}^{\infty} \frac{2 H_{k} z^{k+2}}{(k+1)(k+2)^{j}}
\end{equation}
Where
\begin{equation}
H_{k} = \sum_{n=1}^{...
- 87
2
votes
1
answer
138
views
Powers of polylogarithms
I would like to take powers of arbitrary order to polylogarithm functions. For instance, given
$$
\text{Li}_\alpha(z) = \sum_{k=1}^\infty \frac{z^k}{k^\alpha}
$$
I am interested in
$$
[\text{Li}_\...
- 541
2
votes
0
answers
154
views
Evaluating a variant of the polylogarithm
Consider the infinite sum :
$$\sum_{m=1}^{\infty}\binom{m}{my}\frac{z^{m}}{m^{s}}\;\;\;\;s\in\mathbb{C},\;\;\;\;|z|<\frac{1}{2},\;\;\;\;\;0<y<1$$
I want to evaluate this summation in terms ...
5
votes
3
answers
268
views
On the series expansion of $\frac{\operatorname{Li}_3(-x)}{1+x}$ and its usage
Recently I have asked about the evaluation of an integral involving a Trilogarithm $($which can be found here$)$. Pisco provided a quite an elegant approach starting with a functional equation of the ...
- 16.2k
3
votes
2
answers
185
views
Simple way to evaluate $f(n)= \sum_{r=0}^{\infty}r^n x^r$
Does anyone know a simpler formula than the one below for calculating values of this function for any positive integer n?
$$f(n)= \sum_{r=0}^{\infty}r^n x^r$$
Here's the derivation for the best ...
user363559
2
votes
0
answers
74
views
How can I prove that the following function is increasing in $x \in [0,1]$?
How can I prove that the following function is increasing in $x$: $$\sum_{i=1}^{\infty} x (1-x) ^ {i-1} \log \left (1+ \mu (1-x)^{i-1} \right)$$
where $\mu$ is any non-negative number and $x$ is in $[...
- 21
1
vote
0
answers
62
views
Convergence of some sums of complex functions
In the context of the probability theory of rare events i found myself dealing with these series of complex functions:
$\sum_{n=1}^\infty(1+n)^{-k}z^{n^2}\\$ with z Complex and k Real.
$\sum_{n=1}^\...
- 11
4
votes
3
answers
170
views
There's a small detail in this proof on why $\sum_{k=1}^{\infty}\frac{1}{k^2} = \frac{\pi^2}{6}$ that I can't figure out
http://www.maa.org/sites/default/files/pdf/upload_library/2/Kalman-2013.pdf
Here is a link to the article I have been reading. It's really interesting and easy to follow. What bothers me is a result ...
- 987