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Results tagged with sequences-and-series
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user 37266
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
0
answers
374
views
Convergence of the series $\sum_{n=1}^\infty \frac{(2+\sin n)^n}{3^n n^a}$ for $a\in(0,1)$
This is inspired by this Math.SE question, for $a=1$.
Borwein, Bailey, and Girgensohn pose in their book ([1,Problem 35]) as an open problem the convergence of the series
$$\sum_{n=1}^\infty \frac{(2+ …
- 1,372
1
vote
0
answers
53
views
Asymptotics of $K$-functional between $\ell_1$ and $\ell_2$ for a specific sequence
I originally had posted this question on Math.SE, two weeks ago. Since it is research-based (even though I am not 100% confident it fits the bill for MathOverflow) and didn't receive any answer on Mat …
- 1,372
11
votes
1
answer
2k
views
Biased random Fibonacci sequences
I have recently been toying (very superficially) with the random Fibonacci sequence, i.e., defined by $F_0=1=F_1=1$ and
$$
F_{n} = F_{n-1} + \varepsilon_n F_{n-2}
$$
where $(\varepsilon_n)_{n\geq 2}$ …
- 1,372
2
votes
0
answers
183
views
Properties of the optimal decomposition for the $K$-functional between $\ell_1$ and $\ell_2$
Background: For any fixed $t> 0$, the $K$-functional defines a norm on the space $\ell_1+\ell_2$:
$$
\lVert a\rVert_{K(t)} = \inf\{\lVert a'\rVert_1+ t\lVert a''\rVert_2 : a'\in\ell_1,\ a''\in\ell_2,\ …
- 1,372