For questions about recurrence relations, convergence tests, and identifying sequences

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Let $a_{2n-1}=-1/\sqrt{n}$ for $n=1,2,\dots$ Show that $\prod (1+a_n)$ converges but that $\sum a_n$ diverges.

Let $a_{2n-1}=-1/\sqrt{n}$, $a_{2n}=1/\sqrt{n}+1/n$ for $n=1,2,\dots$ Show that $\prod (1+a_n)$ converges but that $\sum a_n$ diverges. What I have found so far is that $\prod_{k=2}^{2n} ...
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0answers
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The recurrence $a(n,k) = \sum_{0\leqslant j<n} a(n+j,k-1)$

I'm trying to find a closed form expression to the following recurrence relation: \begin{align} &a(0,k) = 0, \quad \forall k\geqslant 1; \\ &a(1,k) = 1, \quad \forall k\geqslant 1; \\ ...
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1answer
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Visual proof of $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$?

In a gorgeous paper "How to compute $\sum \frac{1}{n^2}$ by solving triangles", Mikael Passare offers this idea for proving $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$: Proof of equality ...
2
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2answers
45 views
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Integer reciprocals in arithmetic progression

Let $m_1,m_2,\ldots,m_k$ be $k$ positive integers such that their reciprocals are in AP. Show that $k<m_1+2$. Also find such a sequence. Whatever way I tried, whichever formula I used, I ...
7
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0answers
111 views
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Show the equivalence of two infinite series over Bessel functions

The following sums pop up in diffraction theory and are related to Lommel's function of two variables. Let $u,v\in\mathbb{R}$. I claim that $$\sum_{n=0}^\infty i^n \left ( \frac{u}{v} \right )^n ...