0
votes
0answers
49 views

closed form expression for an infinite series

Is there any closed form expression for the infinite sum $∑_{n≥0}q^{n(n+1)/2}(1+q)(1+q^2)...(1+q^n)u^n$ where both q and n are variables and $n \in N∪0$?
5
votes
2answers
73 views

Closed form of generating function consisting of power of two binomials

Let $g(x)$ be infinite formal power series and $$g(x) = (1 + x)(1 + x^2)\cdots(1 + x^{2^k})\cdots$$ Show that $(1 - x) g(x) = 1$. My book gives following proof: Using a fact that $(1 - x^k)(1 + ...
0
votes
1answer
40 views

Close form of a power series starting at $n=2$

This is the power series I am looking at $\sum_{n=2}^{\infty}{n(n-1)z^n}$. I want to find the closed form of this power series. This is my approach, if I divide the power series by $z^2$, then I ...
4
votes
3answers
268 views

Find a closed form from the given power series

I have the power series $\sum_{n=0}^{\infty} {z^{2n}\over{n!}}$, how do I find the closed form for this power series. I am aware that $e^z=\sum_{n=0}^{\infty} {z^{n}\over{n!}}$, so I tried to ...
2
votes
3answers
112 views

Explicit formula for the series $ \sum_{k=1}^\infty \frac{x^k}{k!\cdot k} $

I was wondering if there is an explicit formulation for the series $$ \sum_{k=1}^\infty \frac{x^k}{k!\cdot k} $$ It is evident that the converges for any $x \in \mathbb{R}$. Any ideas on a formula?
1
vote
0answers
76 views

Proof for closed form approximation of e

I am familiar with the derivation of $e$ from a power series, $e = \sum_{k=0}^{\infty} \frac{1}{k!} $ but have not found the proof for the following representation in any textbook $e = \lim_{x\ \to ...
7
votes
1answer
228 views

Closed form for the series $\sum\limits_{k=0}^\infty \frac{(-1)^k\exp(-\lambda(2k+1)^2)}{(2k+1)^3}$

Does there exist an explicit expression for $$\sum_{k=0}^{\infty }\frac{\left( -1\right) ^{k}e^{-\lambda \left( 2k+1\right) ^{2}}}{\left( 2k+1\right) ^{3}}\;,$$ where $\lambda$ is a positive scalar? ...
0
votes
1answer
62 views

What are the applications of being able to find closed forms of “truncated” power series?

Suppose that we have a power series: $$f(x) = \sum_{k=0}^\infty{c_i x^i}$$ Now suppose that we can get a closed form, e.g. a function composed of simple arithmetic of elementary functions, for the ...
3
votes
3answers
555 views

How to get closed form from generating function?

I have this generating function: $$\frac{1}{2}\, \left( {\frac {1}{\sqrt {1-4\,z}}}-1 \right) \left( \,{ \frac {1-\sqrt {1-4\,z}}{2z}}-1 \right)$$ and I know that $\frac {1}{\sqrt {1-4\,z}}$ is ...
2
votes
2answers
304 views

Identify this power series / solve this trig equation

I was asked to find a solution to $$\frac{\sin^2(nx)}{n^2\sin^2(x)}=2^{-1/2}$$ where $n$ is a fixed integer greater than 1. Numerically, there's a solution just above 1/n so I decided to find this ...