# Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: recurrence relations, convergence tests, identifying sequences, identifying terms. For questions on finite sums, use the (summation) tag instead.

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### Recurrence $a_{n}=a_{\lfloor 2n/3\rfloor}+a_{\lfloor n/3\rfloor}$

I am considering the sequence $$a_n=a_{\lfloor 2n/3\rfloor}+a_{\lfloor n/3\rfloor}$$ with $a_0=1$, and I would like to calculate the limit $$\lim_{n\to\infty} \frac{a_n}{n}$$ I have seen this famous ...
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### On $\int_0^\infty \frac{\exp(-x^2)}{1+x^2}dx=\frac{\pi e}2\text{erfc}(1)$

I was attempting to answer this question, but then I came across a question of my own involving my attempt. Task: Prove $$\int_0^\infty\frac{\exp(-x^2)}{1+x^2}\mathrm dx=\frac{\pi e}2\text{erfc}(1)$$ ...
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### Gap in spiral sequence

OEIS sequence A272573 describes a sequence generated in the following way: Start a spiral of numbers on a hexagonal tiling, with the initial hexagon as a(1) = 1. a(n) is the smallest positive ...
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### Determining finitude or infinitude from a simple geometric construction

Playing with a pencil on a checkered sheet I encountered this construction: 1) take a point $A$ on the grid and a point $B$ that is distant from $A$ $n=2,3,4...$ horizontal steps and $1$ vertical ...
### Check simple proof that $\lim\limits_{s\to0^+}\sum\limits_{n=1}^\infty(-1)^nf(n)^{-s}=-\frac12$ if $f>0$, $f''\le0$ and $f(+\infty)=+\infty$
$f:[1,+\infty)\to \mathbb{R}_+$ satisfies $\ f''\leq0,\ f(+\infty)=+\infty$. Prove $$\lim_{s\to0^+}\sum_{n=1}^\infty (-1)^n[f(n)]^{-s}=-\frac{1}{2}$$ Some of my friends showed me this question ...