# Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: recurrence relations, convergence tests, identifying sequences, identifying terms. For questions on finite sums, use the (summation) tag instead.

6,909 questions
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### Sorting of prime gaps

Let $g_i$ be the $i^{th}$ prime gap $p_{i+1}-p_i.$ If we re-arrange the sequence $(g_{n,i})_{i=1}^n$ so that for any finite $n$ the gaps are arranged from smallest to largest we have a new sequence ...
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### A Ramanujan-like summation: is it correct? Is it extensible?

I'm still exercising with summation-procedures which I try to make correct Ramanujan-summations. I'm looking at the (gap-)series $$s(1/2,2) = (1/2)^1+(1/2)^{4}+(1/2)^{9}+(1/2)^{16}+(1/2)^{25}+...$$ ...
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### Have I discovered an analytic function allowing quick factorization?

So I have this apparently smooth, parametrized function: The function has a single parameter $m$ and approaches infinity at every $x$ that divides $m$. It is then defined for real $x$ apart from ...
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### Asymptotic related to the infinite product of sine

The amount is somewhat complicated ($x$ is a constant): $$S_n=\sum_{k=1}^n\ln\left(1-\frac{\sin^2\big(x/(2n+1)\big)}{\sin^2\big(k\pi/(2n+1)\big)}\right)\tag{*}$$ I want to enrich my handy powerful ...
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### On the weak and strong convergence of an iterative sequence

I have some difficulties in the following problem. I would like to thank for all kind help and construction. Let $H$ be an infinite dimensional real Hilbert space and $F: H\rightarrow H$ be a ...
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### How much can we rearrange a series?

There's a well-known result that if $\sum a_n$ is conditionally convergent, then for any real $c$ there exists a permutation $\pi:\mathbb{N} \to \mathbb{N}$ such that $\sum a_{\pi(n)} = c$. A ...
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### Number as the sum of digits of some degree

We will say that the measure of a number is equal to the maximum degree in which it is possible to represent a number in the form of a sum of digits copied (You can not rearrange the numbers). For ...
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### Have “groupy” numbers been studied before?

In number theory, a positive integer $n$ is called highly composite if it has more divisors than any smaller positive integer. This notion has been studied by several notable mathematicians; for ...
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### How to sum up this series and simplify yet another one?

Primarily, I would like to know what could be done with this series: $$\sum_{n=2}^{\infty}\frac{n^3}{(n^2-1)^3}\left(\frac{n-1}{n+1}\right)^{2n}$$ As hardmath says in his comment, the series ...