# 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”?

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### Spherical harmonic expansion of a power function

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### An inequality for a recursively defined sequence of numbers

**2**

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### Truncated Euler products, Dirichlet eta function, and convergence issues

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### Formulas for $\sum_{x=1}^{n}\Big\{\frac{x^q}{n}\Big\}$

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### The sequence $a(n)=(2^n \bmod p)^{p-1} \bmod p^2$

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### If $P_n \rightrightarrows P$ in $\mathbb{R}$ and $P_n$ are polynomials proof that $P$ is polynomial [closed]

**4**

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### A problem on rate of decay of fill distance?

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### Is the arithmetic-geometric mean of 1 and 2 rational?

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### Can I write this series in a recursive way?

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### Positivity conjecture for Somos sequences

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### A closed form or a good approximation of an infinite series related to the negative binomial distribution

**-5**

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### What is the pattern in this sequence of fractions? [closed]

**10**

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### What can one say about $\sum\limits_{i=1}^\infty \frac{1}{p_{i+1}^2-p_i^2}$?

**14**

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### Calculation of a series

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### What is the regularized numerocity of prime numbers?

**2**

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### Tail asymptotics of Durfee square identity

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### Convergence mode with inputs and functions varying in tandem

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### Sequences over finite fields

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### Asymptotics of a combinatorial series

**2**

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### sum of odious numbers to the power of k

**5**

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### Validity of $\ln z=\frac{\pi}{\operatorname{AGP}(\theta_2^2(1/z),\theta_3^2(1/z))}$

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### Is this pair of coupled sequences known, and what are their properties?

**2**

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### Banach limit with added properties

**7**

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### Fraction of elements in $\mathbb{Z}_n$ satisfying a certain equation

**1**

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### Comparing growth of sequences in weighted spaces

**7**

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### Comparing divergent and convergent sums

**5**

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### Is the harmonic series worse than any summable series?

**4**

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### Is there a conjecture about the bounds (constant or a function) of $\sum_{n \le x} \mu(n)/\sqrt{n}$

**10**

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### When exactly is the principal AGM equal to the optimal AGM?

**1**

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### A dig at Ramanujan's: $\sum_{k=1}^{\infty} (-1)^{k-1} \frac{x^{pk}}{k(k!)^p} \sim p \ln x +p \gamma,~ p>0$

**3**

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### Finding the summation formula for the recurrence relation $T_n=T_{n-1}\;+(n-1)\cdot T_{n-2}\;$

**2**

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### Logistic sequence convergence

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### Minimal growth condition for a rearrangement

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### Log-convexity of Lassalle's sequence

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### Organization Program of Random Integers

**9**

**3**answers

### Method to evaluate an infinite sum of ratio of Gamma functions (how does Mathematica do it?)

**1**

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### Convergence of numerical method [closed]

**10**

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### Asymptotic analysis of $x_{n+1} = \frac{x_n}{n^2} + \frac{n^2}{x_n} + 2$

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### Applying Tannery's theorem to generalised hypergeometric functions

**2**

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### Evaluations of three series involving binomial coefficients

**2**

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### Find better than $ 4^n\prod_{k=1}^{n-1}\cos^2(k)\sim e^{o(n)}$

**1**

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### Prove that: $\sum _{c=1}^n \sum _{b=1}^n \sum _{a=1}^n \left(\left([b|c][b|a]\frac{\mu(b)b}{a}\right)-\frac{1}{a b\sqrt{c}}\right)<H_n+n$

**2**

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### About $\lim_{n\to +\infty} n\prod_{k=1}^{n-1}\cos^2(k)$

**2**

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### Are there theorems dealing with the “amount of oscillatory divergence” of series?

**9**

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### Two-term recurrence relation

**2**

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### How can collections of rational zeta series that are equal to $\sum_{n=2}^{\infty} (\zeta(n) - 1)^{p} $ be obtained?

**11**

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### Strange behavior of $x_{n+1}=x_n +\lambda \sin x_n$

**12**

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### Ordered Bell numbers

**3**

**1**answer