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Results tagged with sequences-and-series
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user 104330
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
Boundedness from convergence of some demeaned sequence
While writing this answer it appears to me that I discovered some sort of a tautology, basically because $a_n$ are uniquely determined by $b_n$. Specifically, by simple algebra we have $\bar{a}_n - \b …
- 6,476
2
votes
Positive linear recurrent sequence
Even more is true: for all linear recurrences either $|a_k| \le xr^k$ for some $x>0, r < 1$ or $\limsup |a_k| > 0$.
Indeed, for any linear recurrence we have $a_k = \sum_{m = 1}^N b_m k^{c_m} d_m^k$ f …
- 6,476
5
votes
Accepted
Does $\sum_{n=1}^\infty e^{-n^2 T} \int_0^T e^{n^2 t} \lvert f(t)\rvert \, dt$ converge for ...
The updated version is indeed correct. I will show that the sum is finite for almost all $T\in [1, 2]$ but it should be applicable for all $T\in (0, \infty)$.
The reason why the result can fail for a …
- 6,476
3
votes
Is this closed-form summation a special case of known Lerch zeta function formulas?
Let me suggest a more pedestrian approach than the elegant solution of Christian Remling but which is arguably easier to come up with: contour integration.
To get the sum over $\mathbb{Z}$ from a cont …
- 6,476
1
vote
Accepted
Lower bound in recurrence relation
This recurrence looks fairly random, so let's try tackling it in parts. First, we will show that $\log v_k$ and $\log n_k$ grow like $k^2$. Indeed, it can not grow slower since even if we leave the fi …
- 6,476
3
votes
Accepted
About the recursive inequality $w_p \geq (1-\frac {\pi}n)w_{p-2n} + 2\pi + o(1)$
I will show that $a \ge \sqrt{\pi}$ which I assume is what you wanted to ask (and it complements counterexample of Iosif Pinelis which appeared while I was writing this answer, showing that this is sh …
- 6,476
14
votes
On the finite sum of reciprocal Fibonacci sequences
Let me sketch a proof that this identity holds for big enough $n$. In fact, we can show that
$$\left(\sum_{k = n}^{2n} \frac{1}{F_{2k}}\right)^{-1} = F_{2n-1} + \frac{1}{\varphi \sqrt{5}} + o(1),$$
wh …
- 6,476