# Questions tagged [sequences-and-series]

for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.

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### Did anyone ever propose the distinction between “divergent to infinity” as opposed to “divergent but with finite average”?

There are different regularization methods that allow us to ascribe finite values to divergent integrals, series or sequences. Still, in my view there is fundamental difference between divergent ...
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### Spherical harmonic expansion of a power function

Let $f$ be an even continuous function on the sphere $S^{n-1}$. Find a relation of the spherical harmonic expansion between coefficients of $f^n$ and those of $f$.
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### Formulas for $\sum_{x=1}^{n}\Big\{\frac{x^q}{n}\Big\}$

Consider sum: \begin{equation} S_q(n) = \sum_{x=1}^{n}\Big\{ \frac{x^q}{n} \Big\} \end{equation} where $\{x\}$ is fractional part of $x$. It's easy to see that $S_{1}(n) = \frac{1}{2}(n-1)$, but ...
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### A closed form or a good approximation of an infinite series related to the negative binomial distribution

Does anyone know a closed form for this expression: $$\sum_{r =1}^{\infty}{{\alpha + 2r - 1}\choose{ r - 1}}(1 - p)^{\alpha + r}p^{r},$$ where $\alpha \geq 1$ and $0<p<1$. A good approximation ...
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### What is the pattern in this sequence of fractions? [closed]

1/2, 1/2, 5/8, 5/7, 17/22, 13/16,... I notice the top numbers are all primes but could not find how that helps. At first I thought maybe it is similar to a Fibonnaci type sequence because of the first ...
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### What can one say about $\sum\limits_{i=1}^\infty \frac{1}{p_{i+1}^2-p_i^2}$?

Denoting by $p_i$ the $i$-th prime, is it known that $\displaystyle \sum_{i=1}^\infty \frac{1}{p_{i+1}^2-p_i^2}$ converges? Can one compute a few digits based on euristic considerations or plausible ...
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### Calculation of a series

It seems that we have: $$\sum_{n\geq 1} \frac{2^n}{3^{2^{n-1}}+1}=1.$$ Please, how can one prove it?
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### What is the regularized numerocity of prime numbers?

Basically, I want to find this sum: $$\lim_{s\to0}\sum_{n=1}^\infty \frac{p(n)}{n^s}$$ where $p(n)$ is the membership function of the set of primes. That is, $p(n)=1$ if $n$ is prime and $0$ otherwise....
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### Tail asymptotics of Durfee square identity

This post is related to the problem Asymptotics of a combinatorial series According to the Durfee square identity: $$\sum_{k \ge 0} \frac{q^{k^2}}{(q;q)_k^2} (q;q)_{\infty} = 1,$$ where $(q;q)_k$ is ...
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### Is this pair of coupled sequences known, and what are their properties?

I was examining the following pair of 'coupled' sequences (I don't know the correct terminology): $a_{n+1}=a_n+b_n+\frac{a_n}{b_n}$ $b_{n+1}=b_n\left(1+\frac{b_n}{a_n}\right)$ Both sequences grow ...
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### Banach limit with added properties

Let $c=\{a:\mathbb{N}\rightarrow \mathbb{C}: \exists \alpha\in \mathbb{C} \textrm{ so that } \lim\limits_{n\rightarrow \infty} a(n)=\alpha=:L_c(a)\}\subset \ell^\infty$, where $\ell^\infty$ is the ...
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### Fraction of elements in $\mathbb{Z}_n$ satisfying a certain equation

From a question arising in Game Theory, I want to calculate the sequence $$a_n = \max_{f_A, f_B : \mathbb{Z}_n \to \mathbb{Z}_n} \frac{\# \left\{ (x,y) | f_A(x) - f_B(y) = xy \mod n \right\}}{n^2}$$...
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### Finding the summation formula for the recurrence relation $T_n=T_{n-1}\;+(n-1)\cdot T_{n-2}\;$

The exponential generating function of this recurrence relation, $T_n=T_{n-1}\;+(n-1)\cdot T_{n-2}\;$, is $$f(x)=e^{x + \frac{x^2}{2}}$$ Multiplying the exponential generating functions for each term, ...
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### Organization Program of Random Integers

Suppose you have an infinite list of distinct positive integers, $x_1, x_2, \ldots$ and you don't know what each $x_i$ is, for any $i$. However, you are allowed to "compare" any two of the ...
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### Strange behavior of $x_{n+1}=x_n +\lambda \sin x_n$

Consider a sequence $(x_n)$ satisfying $x_{n+1}=x_n +\lambda \sin x_n$. You would expect the sequence $x_n$ to depend on $x_0$ and to exhibit a chaotic, Brownian-type behavior, and indeed it does ...
### Integrality of ratios of $\ell$-sequences
The so-called $\ell$-sequences are defined by $a_0=0, a_1=1$ and $a_n=\ell\,a_{n-1}-a_{n-2}$. The Generalized Lecture Hall Theorem (due to Mireille BousquetMelou and Kimmo Eriksson) depends on a ...