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# Questions tagged [cesaro-summable]

For questions about Cesàro summation and Cesàro summable sequences.

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### Showing a positive Cesaro summable series is normally summable.

Question (Spivak, 23-13): Suppose that $a_n>0$ and $\{a_n\}$ is Cesaro summable. Suppose also that the sequence $\{na_n\}$ is bounded. Prove that the series $\sum_{n=1}^{\infty}a_n$ converges. My ...
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### Relation between the sets of almost convergent, bounded statistical convergent and Cesaro convergent sequences

$\bullet$ A sequence $x=(x_n)$ is said to be Cesaro-summable or Cesaro-convergent to $l$ if the sequence $y=(y_n)$ defined by $y_n=\frac{x_1+x_2+x_3+\dots+x_n}{n}$, converges to $l$. $\bullet$ A ...
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### General Cesaro summation with weight

Assume that $a_n\to \ell$ is a convergent sequence of complex numbers and $\{\lambda_n\}$ is a sequence of positive real numbers such that $\sum\limits_{k=0}^{\infty}\lambda_k = \infty$ Then, show ...
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### $a_n=(-1)^{n-1}, \; s_n=\sum_{i=1}^{n}a_i$ then find $\lim_{n\to \infty}\frac{s_1+s_2+\dots s_n}{n}$

$a_n=(-1)^{n-1}, \; s_n=\sum_{i=1}^{n}a_i$ then find $\lim_{n\to > \infty}\frac{s_1+s_2+\dots s_n}{n}$ $$s_k=1,\; \text{if k is odd and } s_k=0 \text{ if k is even}$$ Cauchy's theorem for a ...
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### Convergence of infinite products

I wonder, parallel to the theory of summability of infinite series is there a theory for infinite products? Is there any generalized convergence method (such as Cesaro and Abel summability) for the ...
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### Cesàro summability

Suppose $(a_n)$ is a Cesàro summable sequence of positive real numbers (i.e., $\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^n a_i$ exists and is finite) and $(b_n)$ is a bounded sequence of positive ...
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### Banach limit for Cesaro summable sequences

I'm solving an exercise from Lax's Functional analysis. The section concerns generalized limits (more particularly, Banach limits), which are obtained by applying the Hahn-Banach theorem to the ...
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### Hardy's power series. Cesàro convergence

Doing my analysis homework i have come across the following power series known as Hardy's power series $$\sum\limits_{k=0}^{\infty}a_kx^k=\sum\limits_{k=0}^{\infty}(-1)^kx^{2^k}\mbox{ for x}\in[0,1],$$...
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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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### Summability Question

Today I was reading Cesaro Summability and Abel summability. I found that there exists a series which is Cesaro summable but do not converge in conventional way (the usual way...). Again there exists ...
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### Summarise with Cesaro summation [closed]

Lets consider series $A = \sum_{n=0}^\infty(-1)^n$. $A = 1 - 1 + 1 - 1 \dots$ Lets consider k-th term of series A. Move it to $2^kth$ position. So, and repeat it for every term in series. Obviously, ...
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### Show a series is Cesaro summable.

I am given this series: $\sum_{n=1}^{\infty}\cos(\frac{n\pi}{6})$ and asked if it converges and if it's Cesaro summable or not. I can easily show that this series diverges. However, I am unsure how ...
(I'm sorry if someone has already asked a similar question, I couldn't find anything from my search). The question is here. Let $l^1$ denote the space of all absolutely summable sequences, i.e., ...