# Tagged Questions

**2**

votes

**0**answers

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

**3**answers

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

**1**answer

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

**2**answers

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

**2**answers

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

**4**answers

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

**2**answers

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 = ...