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Results tagged with sequences-and-series
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user 169474
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.
10
votes
Solving a limit about sum of series
The function $x \mapsto t^{x^2}$ decreases on $\mathbb{R}_+$, so for every $n \in \mathbb{N}$,
$$t^{(n+1)^2} \le \int_n^{n+1}t^{x^2}dx \le t^{n^2}$$
By summation over $n$,
$$\sum_{n=1}^{\infty}t^{n^2} …
- 3,808
2
votes
Does there exist a continuous function $f(x)$ such that $f(0)=0$ and $0<\lim_{n\to\infty}\pr...
Possible way to find such a function $f$
Since
$$\prod_{k=1}^n \frac{ke}{n} = n!\Big(\frac{e}{n}\Big)^n \sim \sqrt{2\pi n} \text{ as } n \to +\infty,$$
it is sufficient to find a continuous function $ …
- 3,808
1
vote
Accepted
Convergence and roots of alternating periodic infinite series
I prove the convergence of the series.
For $n \ge 1$, let
$$S_n = \sum_{k=1}^n k^{-i\beta-\alpha} \text{ and } S'_n = \sum_{k=1}^n (-1)^ {k-1} k^{-i\beta-\alpha}$$
Then
$$S_{2n}-S'_{2n} = 2 \sum_{k=1} …
- 3,808