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Results tagged with sequences-and-series
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user 75
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.
3
votes
Accepted
Sets of natural numbers with finite intersections and divergent sums of reciprocals
This can be done by a fairly simple diagonalization argument. Call a subset of $\mathbb{N}$ large if its sum of reciprocals is infinite. It suffices to prove the following (for we can then construct …
- 31.2k
28
votes
Accepted
Which sequences can be extended to analytic functions? (e. g., Ackermann's function)
It's a standard theorem in complex analysis that if $z_n$ is a sequence that goes to infinity, there is an entire function taking any prescribed values at the $z_n$. There is a function $f$ vanishing …
- 31.2k
12
votes
Is this a rational function?
Write $f_0(z)=\sum\frac{z^n}{2^n-1}$ and inductively define $f_k(z)=f_{k-1}(z)-\frac{1}{1-z/2^k}$. By induction, it is easy to see that $f_k$ is given by the power series $$f_k(z)=\sum\frac{z^n}{2^{k …
- 31.2k