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Questions tagged [cesaro-summable]

For questions about Cesàro summation and Cesàro summable sequences.

If $\{a_k\}_{k = 1}^{\infty}$ is a sequence, define a sequence of partial sums $s_k = a_1 + a_2 + ... + a_k$. We call the sequence $$\frac{s_k}{k} = \frac{a_1 + ... + a_k}{k}$$ the Cesàro means of the sequence.

The sequence $\{a_k\}$ is called Cesàro summable if the sequence of Cesàro means converges. If the original sequence $\{a_k\}$ converges to some $A$, then the sequence of Cesàro means also converges to $A$. But the converse does not hold: There are divergent sequences whose Cesàro means converge (e.g. $a_n = (-1)^n$).

Reference: Cesàro summation.

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

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…
Mykolas
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Convergence of series implies convergence of Cesaro Mean.

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 - c_1(N-1)/N + c_2(N-2)N +\dots+c_{N-1}/N \\…
Daniel Donnelly
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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…
NessunDorma
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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 sufficiently rigorous. Can the proof be repaired to…
ziggurism
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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 $\ell_\infty$? As usually, $\ell_\infty$ denotes the…
Martin Sleziak
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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 $$\lim_{M\rightarrow\infty} \frac{\sum_{k=1}^M w_k}{M}$$ does not exist? I…
tvk
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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 zero? As in $$ \lim_{n\to\infty} \frac{1}{n}…
Daniel Mansfield
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$u_n$ converges if and only if $\frac1{\sum_{k=0}^n a_k} \sum_{k=0}^n a_ku_k $ converges.

Let $(a_n)_{n \in \mathbb{N}} $ be a positive sequence with $a_0\neq0$. Find a necessary and sufficient condition on $(a_n) $ in order that: for any real sequence $(u_n)_{n \in \mathbb{N}}$, $$\boxed{(u_n)_{n \in \mathbb{N}}\quad…
Paul
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Follow-up Question about Cesaro mean proof

I am trying to understand the proof behind the Cesaro mean converging. I am using https://math.stackexchange.com/a/2342856/633922 (hopefully it is also correct) as a guide because it seems very direct. I will comment on the steps I understand and…
Mobius.Drip
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Is the product of two Cesaro convergent series Cesaro convergent?

Let $\{a_n \}_{n \geq 1}$ and $\{b_n \}_{n \geq 1}$ be two sequences of real numbers such that the infinite series $\sum\limits_{n=1}^{\infty} a_n$ and $\sum\limits_{n=1}^{\infty} b_n$ are both convergent in the Cesaro sense i.e. \begin{align*}…
ACB
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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 classical limit functional on $\ell^\infty$. The…
MSDG
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(Non-) Convergence of $\frac{1}{n} \sum_{k=0}^{n - 1} \exp\left(2i \pi [\frac{3 + \sqrt{5}}{2}]^k\right)$ when $n \to +\infty$

Let be $$\forall n > 0, S_n = \dfrac{1}{n} \sum\limits_{k=0}^{n - 1} \exp(2i\pi u_k),\quad \forall k \geq 0, u_k = \left(\dfrac{3 + \sqrt{5}}{2}\right)^k$$ I would like to prove or disprove the convergence of $S_n$ as $n \to +\infty$. What I have…
Raito
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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: $$ \begin{split} \lim_{n \to \infty}\frac{1}{n}\sum_{k=0}^{n-1} s_k &= A \implies \lim_{t \to…
AlexInorbit
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A result on sequences: $x_n\to x$ implies $\frac{x_1+\dots+x_n}n\to x$ without using Stolz-Cesaro

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 x_n \le \liminf \frac{\sum_{i=1}^{n} x_i}{n} \le…
user41281
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Is the Cesàro sum of the Fourier series of $f$ a best approximation of $f$ in any sense?

Let $(e_n)_{n\in\mathbb{\mathbb{Z}}} = (t\mapsto e^{int})_{n \in \mathbb{Z}}$ the Fourier orthonormal base of $L^2(\mathbb{T})$ with the scalar product $\langle f,g\rangle = \int_{-\pi}^\pi f(t)\bar{g}(t)\frac{\operatorname{d}t}{2\pi}$. Define for…
Bob
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