2
votes
0answers
161 views

closed form expression for an infinite series

Is there any closed form expression for the infinite sum $\sum_{n \geq 0}q^{n(n+1)/2}(1+q)(1+q^2)\cdots(1+q^n)u^n$ where both $q$ and $n$ are variables and $n \in N \cup {0}$?
4
votes
3answers
292 views

Sum of series $a^{i^2}$

Is there any closed form known for the expression $\sum_{i=1}^\infty a^{i^2}$ where $|a|<1$? Thanks!
21
votes
1answer
290 views

Rearrangements of a power series at the boundary of convergence

Take some power series $f(z) = \sum a_n z^n$ with a finite non-zero radius of convergence. I can rearrange the terms of the series, say, to get a different infinite series $$f_{\sigma}(z) = ...
2
votes
2answers
328 views

Inverting an asymptotic series

I have the first few terms of a series of the form, $y(x)=\ln(x)+x+a_0+\frac{a_1}{x}+\frac{a_2}{x^2}+\cdots$. Knowing that the inverse $x(y)$ exists, I am looking for method to write x in terms of ...
8
votes
2answers
795 views

What is the series expression for (1+1/x)^x about x = \infty?

This seems like it must have been addressed somewhere already, but I cannot find it in any standard series tables. I have the equation: $f(z) = \left(1 + \frac{1}{z}\right)^z$. What is the general ...
2
votes
4answers
982 views

Closed-form for modified formal power series

This question have been driving me crazy for months now. This comes from work on multiple integrals and convolutions but is phrased in terms of formal power series. We start with a formal power ...
4
votes
2answers
2k views

power series of the reciprocal… does a recursive formula exist for the coefficients. [closed]

Hello If $f(x)=\sum _{n=0}^{\infty } b_nx^n$, and $\frac{1}{f(x)}=\sum _{n=0}^{\infty } d_nx^n$. Then the coefficients of the reprical of f can be written down. The first few terms are: $d_0 = ...