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Results tagged with sequences-and-series
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user 1306
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.
13
votes
Accepted
Conjectured relation between alternating Prime zeta series and Riemann zeta
We have $$\sum_k\frac{(-1)^kP(nk)}{k}=\sum_{k,p}\frac{(-1)^k}{kp^{nk}}=-\sum_p\ln\left(1+\frac{1}{p^n}\right)=\sum_p\ln\left(\frac{1-\frac{1}{p^{n}}}{1-\frac{1}{p^{2n}}}\right)=\ln\frac{\zeta(2n)}{\ze …
- 16.9k
5
votes
An elementary proof for a limit?
Adapting Fedor's argument, we have
$$
a_n-a_{n-1}=\frac{1}{n\log(n)}+\log\log(n-1)-\log\log n=
\frac{1}{n\log(n)}+\log\left(\frac{\log(n-1)}{\log n}\right),
$$
and, similarly to the calculation from …
- 16.9k
5
votes
Accepted
Lambert series identity
It's just about using geometric series a lot. Indeed, we have
$\sum\limits_{n=1}^\infty\frac{q^n x^{n^2}}{1-qx^n}=\sum\limits_{n=1}^\infty\sum\limits_{k=0}^\infty q^{n+k}x^{n(n+k)}=\sum\limits_{m=1}^ …
- 16.9k
2
votes
0
answers
31
views
joint rank sequences
An algebraic question I have been working on led me to a sequence that appears in OEIS as A186355: "adjusted joint rank sequence of $(f(i))$ and $(g(j))$ with $f(i)$ before $g(j)$ when $f(i)=g(j)$, wh …
- 16.9k