# Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.

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### Sorting of prime gaps

Let $g_i$ be the $i^{th}$ prime gap $p_{i+1}-p_i.$ If we rearrange the sequence $(g_{n,i})_{i=1}^n$ so that for any finite $n$, if the gaps are arranged from smallest to largest, we have a new ...
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### Evaluating $\sum\limits_{x=2}^\infty \frac{1}{!x}$ in exact form.

$$\large \text{Introduction:}$$ We know that: $$\sum_{x=0}^\infty \frac{1}{x!}=e$$ But what if we replaced $x!$ with $!x$ also called the subfactorial function also called the $x$th derangement number?...
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### $2^n$th decimal place of $\sqrt{2}.$

Someone on Art of Problem Solving claims to know how to calculate the $2^{2020}$th decimal place of $\sqrt{2},$ and will tell us if everyone gives up. Brute force will not work, nor will a BBP style ...
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### Combinatorial Proofs of Real Analysis Identity

In this question, a proof using real analysis is given of the following identity: $$\sum_{n=1}^{\infty} \frac{(n-1)!}{n \prod_{i=1}^{n} (a+i)} = \sum_{k=1}^{\infty} \frac{1}{(a+k)^2}$$ Is there a ...
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### A curious coincidence in the series representation of $\zeta(7)$

Let $\zeta(n)$ denote the Riemann Zeta function defined for positive integers $n$ as usual by: $$\zeta(n)=\sum_{m=1}^{\infty} \frac{1}{m^n}.$$ It is currently unknown whether there exists a series ...
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### Does the sequence $x_0=12$ , $x_{n+1}=x_n^2+1$ contain a prime?

I wonder whether the sequence defined by $$x_0=12$$ $$x_{n+1}=x_n^2+1$$ for all non-negative integers $n$ contains a prime number. The following table shows from left to right : The index $n$ , the ...
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### $\int_0^\infty \log\left(1 - \frac{x}{\sinh x}\right)\,dx$ and generalisations

I'm interested in the value of $$\theta(2) := -\frac{1}{2\pi^2}\int_0^\infty \log\left(1 - \frac{x}{\sinh x}\right)\,dx.$$ I have some hopes this admits a closed-form solution. Expanding the ...
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### If $a_n \rightarrow a$ Then $\frac{1}{n} \sum_{k=1}^{n} a_k \rightarrow a$

If $a_n \rightarrow a$ as $n \rightarrow \infty$, then can we say that $\frac{1}{n} \sum_{k=1}^{n} a_k \rightarrow a$ as $n \rightarrow \infty$? If we consider $\frac{1}{n} \sum_{k=1}^{n} a_k$ as ...
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