# Tagged Questions

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### A closed form of $\sum_{k=0}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$

I am looking for a closed form of the following series $$\mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$$ I have no idea how to ...
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### Find whether the following series converges or diverges $\sum_{n=1}^{\infty}\frac{\ln n }{\sqrt{n}}$

Looking for a witty answer. I can see that the given series converges by AST. All Ideas Appreciated
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### Matrix Inversion Test ( Sum of Matrix series)

Friends,I have a set of matrices of dimension $3\times3$ called $A_i$. , Following are the given conditions a) each $A_i$ is non invertible except $A_0$ because their determinant is zero. b) ...
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### Sum of nth powers and generalized polynomial sum

So this is a 2-part question (both parts I believe are closely related): How exactly does on express the sum $$\sum_{i=0}^{k}{i^n} = Q(n,k)$$ in a closed form For arbitrary positive integers ...
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### Intuitive ways to get formula of cubic sum

Is there an intuitive way to get cubic sum? From this post: combination of quadratic and cubic series and Wikipedia: Faulhaber formula, I get $$1^3 + 2^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$$ I think ...
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### Prove the sum $\sum_{n=1}^\infty \frac{\arctan{n}}{n}$ diverges.

I must prove, that sum diverges, but... $$\sum_{n=1}^\infty \frac{\arctan{n}}{n}$$ $$\lim_{n \to \infty} \frac{\arctan{n}}{n} = \frac{\pi/2}{\infty} = 0$$ \lim_{n \to \infty} \frac{ ...
### Find a closed form for $\sum_{k=0}^{n} k^3$ [duplicate]
Find a closed form for $\sum_{k=0}^{n} k^3$. I would appreciate ideas for approaching questions like this in general as well. Thanks.