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Results tagged with sequences-and-series
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user 2480
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.
2
votes
Accepted
Finding closed form of recurrence relation in two variables
$F(n,m)=D(n,m-1)-D(n-1,m-1)$ where $D(n,m)$ are the Delannoy numbers.
- 5,103
5
votes
Is there a nonzero solution to this infinite system of congruences?
$u_n=s_na + t_nb + s_{n+1}c$ satisfies the same recurrence relation as $s_n$ and $t_n$: $u_n = u_{n-1} +2u_{n-2} + 4u_{n-3}$. The question is whether $2^{n+1}\mid u_n$.
Since $v_n=u_n/2^{n+1}$ satisfi …
- 5,103
7
votes
If $x_{n+1}= \frac{nx_{n}^2+1}{n+1}$ then $x_{n}=1$
I suspect the answer is no. First rewrite $x_n=y_n+1$, then the recursion becomes
$(n+1)y_{n+1}=ny_n(y_n+2)=(y_n+2)(y_{n-1}+2)\cdots (y_2+2) (y_1+2)y_1$
and for the integrality of $y_{n+1}$ it is su …
- 5,103
11
votes
2
answers
726
views
What can one say about $\sum\limits_{i=1}^\infty \frac{1}{p_{i+1}^2-p_i^2}$?
Denoting by $p_i$ the $i$-th prime, is it known that $\displaystyle \sum_{i=1}^\infty \frac{1}{p_{i+1}^2-p_i^2}$ converges?
Can one compute a few digits based on euristic considerations or plausible c …
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6
votes
1
answer
223
views
Asympotic density of a very simple sequence
Let $A=\{mn(m+n)\mid n,m\in \mathbb{N}_0\}$. Sorted, this is OEIS sequence A088915. What is its asymptotic behavior? It seems approximately $a(n)=O(n^{1.5})$, but not quite.
I'm actually even more in …
- 5,103
0
votes
0
answers
356
views
Is $\lim_{x\to\infty}\sum_{n=-\infty}^{\infty}\frac{x}{n^2+x^2}=\pi$?
It seems that
$$\lim_{x\to\infty}\sum_{n=-\infty}^{\infty}\frac{x}{n^2+x^2}=\pi.$$
But I can't prove it. I cannot prove that the function is decreasing in $x$ either.
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