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Results tagged with sequences-and-series
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user 9550
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.
25
votes
Series whose convergence is not known
Convergence of $\sum_{k=1}^\infty (-1)^k \frac{k}{p_k}$ is unknown, where $p_k$ is k-th prime (Guy 1994, p. 203; Erdős 1998; Finch 2003, according to Eric Weisstein's MathWorld).
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16
votes
What Are Some Naturally-Occurring High-Degree Polynomials?
I was recently amazed by an answer to this question.
I encountered another curious fact while working with hypergeometric functions. The following absolute value of a complex-valued $_4F_3$ function: …
22
votes
Accepted
An elementary expression for $_3F_2(1,1,9/4;2,2;-1)$
Use formula 16.5.2 from DLMF:
$$_3F_2\left(1,1,\frac94;2,2;-1\right)=\int_0^1{_2F_1}\left(1,\frac94;2;-t\right)dt=\frac45\int_0^1\frac{1-(1+t)^{-5/4}}tdt=\\\\\frac{2}{5}\left(8-4\sqrt[4]{8}+\pi-6\ln2+ …
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3
votes
1
answer
241
views
Reconstructing analytic tetration with a complex height from a thinner set of points
This is a follow-up to my previous question An explicit series representation for the analytic tetration with complex height.
Recall the definition $(11)$ from there:
$$t(z) = \sum_{n=0}^\infty \sum_ …
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42
votes
2
answers
2k
views
Numbers that are generic w.r.t. exponentiation
This is a follow-up to my old question Number of distinct values taken by $x\hat{\phantom{\hat{}}}x\hat{\phantom{\hat{}}}\dots\hat{\phantom{\hat{}}}x$ with parentheses inserted in all possible ways.
…
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29
votes
3
answers
3k
views
An explicit series representation for the analytic tetration with complex height
Tetration is the next hyperoperation after more familiar addition, multiplication and exponentiation. It can be seen as a repeated exponentiation, similar to how exponentiation can be seen as a repeat …
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18
votes
3
answers
1k
views
A curious series related to the asymptotic behavior of the tetration
The tetration is denoted $^n a$, where $a$ is called the base and $n$ is called the height, and is defined for $n\in\mathbb N\cup\{-1,\,0\}$ by the recurrence
$$
{^{-1} a} = 0, \quad {^{n+1} a} = a^{\ …
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10
votes
1
answer
750
views
A conjecture about certain values of the Fabius function
The Fabius function is a smooth monotone function $F:[0,1]\to[0,1]$, satisfying functional equations
$$F(0)=0, \quad F(1-x)=1-F(x)\tag1$$
and
$$F'(x) = 2 \,F(2 x) \quad \text{for} \,\, 0<x<1/2.\tag2$$ …
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