Suppose that a sequence $a_{n}$ is Cesaro summable. Prove that $$\lim_{n \to \infty }\frac{a_{n}}{n}=0$$

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 $\ell_\infty$? As usually, $\ell_\infty$ denotes the space of all bounded sequences with the sup-norm $\|x\|=\sup\limits_{n\in\mathbb N} |x_n|$. Closedness of this set was brought up in comments to an […]

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: $\lim_{n \to \infty}\frac{1}{n}\sum_{k=0}^{n-1} s_k = A \Longrightarrow \lim_{t \to \infty}e^{-t}\sum_{n=0}^{\infty}s_n\frac{t^n}{n!} = A$ $\lim_{x \to 1^-}\sum_{n=0}^{\infty}a_nx^n = A \Longrightarrow \lim_{t \to \infty}e^{-t}\sum_{n=0}^{\infty}s_n\frac{t^n}{n!} = A$. Is there a proof of […]

Suppose $a_n$ and $b_n$ to be Cesaro 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 $$ \lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}b_n, $$ do exist. Is the product sequence $c_n=a_nb_n$ always Cesaro summable?

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 + \cdots+x_n}{n} \longrightarrow x $ then $ \{x_n\} \longrightarrow x $? My intuition says that it is true because $ […]

This question already has an answer here: If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $ 2 answers

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 $$\lim_{M\rightarrow\infty} \frac{\sum_{k=1}^M w_k}{M}$$ does not exist? I encountered this problem while studying the limit of average payoffs criterion for ranking payoff sequences in infinitely repeated games.

This question already has an answer here: Prove convergence of the sequence $(z_1+z_2+\cdots + z_n)/n$ of Cesaro means 3 answers

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 sufficiently rigorous. Can the proof be repaired to become rigorously justifiable? If the proof is wrong, why does the result it computes agree with […]

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: $$(z_{1}+z_{2}+\cdots+z_{N-1})+(z_{N}+z_{N+1}+\cdots+z_{n})$$ where $N$ is value of $n$ for which $|A-z_{n}|<\epsilon$ then taking the limit of this sum devided by $n$ , and noting that the second sum is as close as you wish to $nA$ while the […]

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