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Results tagged with sequences-and-series
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user 10846
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.
9
votes
How to calculate the infinite sum of this double series?
Here is a proof based on Hachino's idea. We have agreed it's enough to
prove that
$$S=\sum_{m=1}^\infty\frac{(-1)^m\tanh(m\pi /2)}{m}=\frac{\log 2-\pi}{4}. $$
Plug in the expansion
$$\tanh(m\pi /2)=\ …
- 2,306
44
votes
Accepted
Proof of "Possible new series for $\pi$" without use of physics
I found an elementary proof using partial fractions. This is motivated by Nemo's observation that very similar-looking series appear in the work of Schlosser (Section 7). The matrix inversion used by …
- 2,306
5
votes
Accepted
Is this infinite series related to some well-known special functions?
This is a special case of Ramanujan's ${}_1\psi_1$ summation, which sums the series
$$\sum_{-\infty}^\infty\frac{(a;q)_n}{(b;q)_n}\,x^n,$$
where $(a;q)_n=(1-a)(1-aq)\dotsm (1-a q^{n-1})$. If we let $b …
- 2,306
4
votes
An equality between $\pi$ and $\Gamma$ function
Your series is a special case of Gauss' hypergeometric series. In standard notation, it is
$${}_2F_1\left(\begin{matrix}1/2,1/2\\2\end{matrix};-1\right).$$
In standard tables (e.g. 15.4.26 in https:// …
- 2,306
2
votes
Exotic series for some mathematical constants from String Theory
My paper on these identities is now available, see
https://arxiv.org/abs/2409.06658 .
Your identity III is the case $x_1=x_2=1/4$, $x_3=-1/2$, $\lambda=0$ and $u=1$ of Cor. 5.1. I didn't look at your …
- 2,306
2
votes
Accepted
Modulo $2$ binomial transform of A243499 applied $k$ times
The definition of $a_1$ given in OEIS is based on a bijection between integer partitions and natural numbers. A partition $\lambda_1\geq\lambda_2\geq\dots\geq\lambda_m>0$ with exactly $m$ parts corres …
- 2,306
1
vote
Accepted
Where is the source of the formula $\sum_{j=0}^\infty \bigl(j+\frac{1}{2}\bigr)^{n-1}\frac{2...
I don't have a reference, but it is not hard to prove the result. You want to prove that if
$$\left(\frac{e^x}{2-e^x}\right)^{1/2}=\sum_{n=0}^\infty a_n \frac{x^n}{n!}$$
then
$$a_n=\frac 1\pi\sum_{j=0 …
- 2,306
18
votes
Possible new series for $\pi$
This is an answer to the related post
Proof of "Possible new series for $\pi$" without use of physics
but it seems relevant also for this question. My guess is that this formula is new as it stands, b …
- 2,306