# 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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### Is this long closed form for pi trivial?

With the help of wolfram alpha we got very long closed form for $\pi$ in terms of algebraic numbers, logarithms of algebraic numbers and $cot^{-1},coth^{-1}$ which can be expressed as logarithms. From ...
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### Why mpmath computes $\sum_{n=2}^\infty (-1)^n\log(n)=\log\left(\frac1 2 \sqrt{2} \sqrt{\pi}\right)$

Working with precision 500 decimal digits, mpmath in sage computes: $$\sum_{n=2}^\infty (-1)^n\log(n)=\log\left(\frac1 2 \sqrt{2} \sqrt{\pi}\right).\tag{1}\label{1}$$ We believe the LHS of \eqref{1} ...
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### Surprisingly long closed form for simple series

For natural $A$ define $$f(A)=\sum_{n=1}^\infty \frac{1}{A^n}\left(\frac{1}{An+1}- \frac{1}{An+A-1}\right)$$ $f(A)$ is BBP (Bailey-Borwein-Plouffe) formula and allows digit extraction in base $A$. ...
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### Examples of infinite dimensional involutions

Examples of infinite dimensional involutions I'm looking for more examples of involutions of the type portrayed below, in which two sets of indeterminates (real or complex) each can be transformed ...
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### The Laplace transform and the Lagrange compositional inversion formula

I'm looking for references which derive the Lagrange inversion formula, given below (in bold), for the Taylor series coefficients of the compositional inverse of a function $f$ analytic at the origin ...
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### Given the formula for each specific term of a sequence how do you find the sum of n terms of the sequence? An = 3*(-3)^n-1. Find sum n = 1 to n = 5? [closed]

Which summation properties/rules are used in determining the summation of a finite series ? The given formula an = 3* (-3)^n-1 gives the value for each specific n term, but how to determine the sum ...
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### What's the fastest way to compute $\log n$ for $n>1$?

As it is well known, if $|x|<1$ then we can compute $\log(1+x)$ by the Taylor series $$\log(1+x)=x-\frac{x^2}2+\frac{x^3}3-\cdots.$$ Thus, to compute $\log n$ with $n>1$, we may employ the ...
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### $f' = e^{f^{-1}}$, a third time

I am of the impression the differential equation $f' = e^{f^{-1}}$ was considered on mathoverflow for the first time here: How do i solve this : $\displaystyle \ f'=e^{{f}^{-1}}$? It was found ...
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### Karamata's Abelian/Tauberian Theorem in the complex plane

The following result is well known (a particular case of Karamata's Tauberian Theorem for Power Series in Corollary 1.7.3 of Regular variation by Bingham, Goldie & Teugels): Fix $c, \rho>0$. If ...
1 vote
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### Infinite sum of Laguerre polynomials: is $\sum_{k=0}^{\infty}\frac{n!}{(k+n)!}x^kL_n^k(x)^2=e^x+P(x)$, with $P$ a polynomial of degree $2n-1$?
I have the following expression: $$\sum_{k=0}^{\infty}\frac{n!}{(k+n)!}x^k(L_n^k(x))^2,$$ where $$L_n^k(x)=\sum_{j=0}^n(-1)^j\binom{n+k}{n-j}\frac{x^j}{j!}$$ is the usual associated Laguerre ...