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Questions tagged [cesaro-summable]

For questions about Cesàro summation and Cesàro summable sequences.

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25
votes
3answers
9k views

Prove convergence of the sequence $(z_1+z_2+\cdots + z_n)/n$ of Cesaro means [duplicate]

Prove that if $\lim_{n \to \infty}z_{n}=A$ then: $$\lim_{n \to \infty}\frac{z_{1}+z_{2}+\cdots + z_{n}}{n}=A$$ I was thinking spliting it in: $$(z_{1}+z_{2}+\cdots+z_{N-1})+(z_{N}+z_{N+1}+\cdots+z_{n}...
2
votes
2answers
558 views

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 ...
16
votes
1answer
13k views

Convergence of series implies convergence of Cesaro Mean. [duplicate]

Proof. Let $\sum_{k = 0}^N c_k \rightarrow s$, let $\sigma_N = (S_0 + \dots + S_{N-1})/N$ be the $Nth$ Cesaro sum where $S_K$ is the $Kth$ partial sum of the series. Then $s - \sigma_N \\= s - c_0 - ...
9
votes
2answers
1k views

Bounded sequence with divergent Cesaro means

Is there a bounded real-valued sequence with divergent Cesaro means (i.e. not Cesaro summable)? More specifically, is there a bounded sequence $\{w_k\}\in l^\infty$ such that $$\lim_{M\rightarrow\...
5
votes
2answers
481 views

A result on sequences: $x_n\to x$ implies $\frac{x_1+\dots+x_n}n\to x$ without using Stolz-Cesaro [duplicate]

If $x_n \to x$, how might we prove $$\lim_{n \to \infty} \frac{\sum_{i=1}^{n} x_i}{n} = x$$ Of course, one has $\limsup x_n = \liminf x_n = x$, and thus, using the Stolz-Cesaro theorem: $$\liminf ...
1
vote
4answers
334 views

If $\lim_{n\to\infty}a_{n}=l$, Then prove that $\lim_{n\to\infty}\frac{a_{1}+a_2+\cdot..+a_n}{n}=l$ [duplicate]

Given $a_n$ be a sequence and IF $\lim_{n\to\infty}a_{n}=l$, Then prove that $\lim_{n\to\infty}\frac{a_{1}+a_2+\cdot..+a_n}{n}=l$ I do not know how to do this. Can someone help me with this? Thanks ...
12
votes
1answer
531 views

Can we show that $1+2+3+\dotsb=-\frac{1}{12}$ using only stability or linearity, not both, and without regularizing or specifying a summation method?

Regarding the proof by Tony Padilla and Ed Copeland that $1+2+3+\dotsb=-\frac{1}{12}$ popularized by a recent Numberphile video, most seem to agree that the proof is incorrect, or at least, is not ...
13
votes
3answers
519 views

Is the product of a Cesàro summable sequence of $0$s and $1$s Cesàro summable?

Suppose $a_n$ and $b_n$ to be Cesàro summable sequences of zeros and ones, $a_n\in\{0,1\}$ and $b_n\in\{0,1\}$, i.e. the limits $$ \lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}a_n, $$ and $$ \...
9
votes
2answers
151 views

Are sequences with Cesaro mean a closed subset of $\ell_\infty$?

How can we show that the bounded sequences which are Cesaro summable, i.e., the sequences such that the limit $$\lim\limits_{n\to\infty} \frac{x_1+\dots+x_n}n$$ exists, form a closed subset of $\ell_\...
4
votes
1answer
1k views

convergence of sequence of averages the other way around

In a vector normed space, if $ \{x_n\} \longrightarrow x $ then $ z_n = \dfrac{x_1 + \cdots+x_n}{n} \longrightarrow x $ Is it true the other way arround too? meaning: if $ z_n = \dfrac{x_1 + \...
3
votes
2answers
441 views

Is the Cesàro summation of a sequence divergent to infinity divergent?

More specifically the question is: If I have a sequence $(u_n)_{n\in\mathbb{N}} \subset \mathbb{R}$ that diverges to infinity. Then it's Cesàro summation sequence $(s_n)_{n\in\mathbb{N}}=(\frac{1}{n+1}...
4
votes
1answer
300 views

Abel/Cesaro summable implies Borel summable?

Does Abel or Cesaro summable imply Borel summable for a series? In other words, for a sequence $(a_n)$ and its partial sums $(s_n)$, is it true that: $\lim_{n \to \infty}\frac{1}{n}\sum_{k=0}^{n-1} ...
6
votes
4answers
356 views

Does $\sum_{i=1}^\infty a_i/i < \infty$ imply that $a_i$ has Cesaro mean zero?

If $(a_i)_{i=1}^\infty$ is a sequence of positive real numbers such that: $$ \sum_{i=1}^\infty \frac{a_i}{i} < \infty. $$ Does this mean that the sequence $(a_i)_{i=1}^\infty$ has Cesaro mean ...
3
votes
2answers
365 views

Cesaro mean of Cesaro means

Is it possible to construct a bounded positive sequence $a_i$, ($0 < a_i < K < \infty$) such that the limit of its Cesaro mean does not exist but the limit of the Cesaro mean of its Cesaro ...
2
votes
1answer
181 views

Proving a converse of the Cesaro theorem under extra assumptions

I'm trying to prove that, given $(u_n)_n \in \mathbb{C}^\mathbb{N}$ verifying $ u_{n+1}-u_n =_{n} o(\frac{1}{n})$, the following holds: $$ \lim_{n\to\infty} \frac{u_1+...+u_n}{n} = a \in \mathbb{C} \...
0
votes
2answers
311 views

Suppose that a sequence is Cesaro summable. Prove…

Suppose that a sequence $a_{n}$ is Cesaro summable. Prove that $$\lim_{n \to \infty }\frac{a_{n}}{n}=0$$
3
votes
2answers
218 views

Cesàro summability and $\sum n \lvert a_n\rvert ^2 < \infty$ implies convergence

How can I prove that if $\sum_{n=1}^\infty a_n$ is Cesàro summable and if $\sum_{n=1}^\infty n |a_n|^2 < \infty$, then $\sum_{n=1}^\infty a_n$ converges?
2
votes
1answer
241 views

Rate of convergence of Cesàro means

For a sequence $a_n = O(n^{-1/2})$ as $n\to\infty$, consider the corresponding Cesàro means $b_n = \frac{1}{n} \sum_{j=1}^n a_j$. Is it possible to derive the rate of convergence for the sequence $...
1
vote
2answers
2k views

Understanding Cesaro summation proof

We define: series $\sum_{n=1}^{\infty} a_n$ $s_n=\sum_{k=1}^{n} a_k, n\in \mathbb{N}.$ $\sigma_n=\frac{1}{n}\sum_{k=1}^n s_k$ $s=\lim_{n\to \infty} s_n$ Proposition: If sequence $(s_n)_n$ ...
1
vote
1answer
94 views

Is the Cesaro Operator normal?

The Cesàro operator $T:ℓ_p→ℓ_p$ is defined by $$(Tx)_k=(1/k)\sum_{j=1}^k x_j$$ where $x=(x_j)$. Is this operator normal?