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Results tagged with sequences-and-series
Search options questions only
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user 3106
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
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answers
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Why does this theta function value yield such a good Riemann sum approximation?
Let $T(q)$ denote the third Jacobi theta function at $z=0$; i.e.,
$$T(q) := \sum_{n\in\mathbb{Z}} q^{n^2}.$$
Then for any $x>0$, $T(\exp(-1/x)) \approx \sqrt{\pi x}$. For example, as the entry for A19 …
- 82.7k
71
votes
8
answers
12k
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Possible new series for $\pi$
In a recent (unfortunately over-hyped) preprint by Saha and Sinha, Field theory expansions of string theory amplitudes (arXiv:2401.05733), they present the following series for $\pi$:
$$\pi = 4 + \sum …
- 82.7k
86
votes
4
answers
11k
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Nonexistence of boundary between convergent and divergent series?
The following is a FAQ that I sometimes get asked, and it occurred to me that I do not have an answer that I am completely satisfied with. In Rudin's Principles of Mathematical Analysis, following Th …
- 82.7k
13
votes
5
answers
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Asymptotics of a Bernoulli-number-like function
Tony Lezard asked me the following question which seemed like it should not be too hard but which I did not immediately see how to answer. Define $f(n,k)$ recursively by $f(1,k) = 1$ and
$$f(n,k) = \ …
- 82.7k