# Questions tagged [summability-theory]

For questions about summation/resummation/regularization methods, ways to assign meaningful values to divergent sum. Tauberian theorems for summation methods and other theoretical and applicational results.

17 questions
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### Do the partial sums of a divergent series converge to Cesaro or Abel sums in some metric?

Let $(a_n)$ be a sequence in $\mathbb{R}$, and let $s_n$ be the $n^{th}$ partial sum of the sequence. Then the Cesaro sum of $(a_n)$ is the limit of the average of the first $n$ partial sums as $n$ ...
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### Show that if $\sum_{n=0}^{\infty}a_n^k=\sum_{n=0}^{\infty}b_n^k$ for all $k\in \Bbb N^*$ then $(a_n) = (b_n)$

Let $(a_n),(b_n)$ two summable sequences. Show that if $\sum_{n=0}^{\infty}a_n^k=\sum_{n=0}^{\infty}b_n^k$ for all $k\in \Bbb N^*$ then $(a_n) = (b_n)$ more or less a permutation What I tried: to ...
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### An oscillating series of real numbers which can be resummed to a complex value

Okay so here I go again studying summability theory I was wondering the following problem but first I'll state a few conventions: A series diverges if the partial sums tends to $\pm \infty$, ...
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### A Summability methods which sum the harmonic series

Studying summability theory I've come across many summation methods however by now I know only two not very interesting method which re-sums the harmonic series $\sum_{n=0}^\infty \frac 1{n+1}$ : the ...
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### Is the series corresponding to a summable family absolutely convergent?

Let $I\ne\emptyset$ be a set $E$ be a normed $\mathbb R$-vector space $(x_i)_{i\in I}\subseteq E$ is called summable with sum $x\in E$ $:\Leftrightarrow$ \forall\varepsilon>0:\exists J\subseteq ...
1answer
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### Every linear and regular summation method is also stable

In the theory of divergent series regularization I've came to the following conclusion and I would like to know if my considerations are right or not. First I'll recall some definitions: A summation ...
0answers
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### Triangle inequality and summation methods for divergent series

I'm focusing on summability theory/summation methods/resummation theory and related topics, in other words methods to give meaningful sum to divergent series. First I recall that a method is called ...
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### Silverman-Toeplitz Theorem

Reading this paper On Deferred Statistical Convergence of Sequences by Kucukaslan and Yilmazturk published in KYUNGPOOK Math. J.56(2016), 357-366 I am stuck at theorem $2.2.7$ I get \$\{n^{(1)} ,n^{(2)...
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### Where to find a proof of Silverman-Toeplitz?

I am referring to the theorem which gives a necessary and sufficient condition on a infinite matrix that maps convergent sequences to sequences converging to the same limits. Wiki gives a link to ...