# Tagged Questions

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### 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 $n \in N∪0$?
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### 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? ...
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### 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 ...
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 ...
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 ...