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Questions tagged [cesaro-summable]

For questions about Cesàro summation and Cesàro summable sequences.

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26 views

If I know the limit of Cesaro averages , then can I know limits of uniform Cesaro averages. Details Below

Say that I have a sequence $\{a_{n}\}_{n\in\mathbb{N}}$ $\subset$ [0,1] such that lim$_{N\rightarrow\infty}$ $\frac{1}{N}$ $\sum_{n=1}^{N}$ a$_{n}$ = a. Can I somehow get the value of lim$_{(N-M)\...
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1answer
342 views

A bound on sup-norm of Fourier series

Let $f$ be a Riemann integrable function on $[-\pi,\pi]$ such that $\hat{f}(n)\leq \frac{K}{|n|}$ for some constant $K$, for all $n\neq 0$. Show that $$|S_N(f)|_\infty\leq |f|_\infty+2K$$ Here, $\...
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0answers
15 views

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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1answer
42 views

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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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}...
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2answers
445 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}...
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0answers
26 views

Geometric meaning of the Césaro limit of Geometric sequence on the Torus.

As a motivattion for an introdutory notion in our Ergodic lecuture, we were asked to give a Geometric meaning to the following limit. Let $\lambda\in \Bbb T$ $$\lim_{n\to \infty}\frac1n\sum_{k=0}^{...
2
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2answers
572 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 ...
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1answer
66 views

$a_n=(-1)^{n-1}, \; s_n=\sum_{i=1}^{n}a_i$ 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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1answer
173 views

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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3answers
526 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 $$ \...
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1answer
301 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} ...
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1answer
41 views

Cesaro Means decrease slower than the sequence

For a strictly decreasing sequence of positive real numbers, I want to show that the Cesaro means decrease slower than the sequence itself. In particular, I need $$\dfrac{C_n}{C_{n+1}}<\dfrac{a_n}{...
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1answer
92 views

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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161 views

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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1answer
84 views

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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1answer
105 views

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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2answers
289 views

Cesàro summability implies convergence of (conventional) sum when $\sum_{n=1}^\infty na_n^2<\infty$

Suppose that series $\sum_{n=1}^\infty na_n^2<\infty$ and $\sum_{n=1}^\infty a_n$ is Cesàro summable. How do you show that $\sum_{n=1}^\infty a_n$ converges? I know that if $\lim_{n\rightarrow \...
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0answers
50 views

Infinite/Recursive Cesàro Summation of $\zeta(1)$

Is anything known about this kind of `infinite' Cesàro summation (or any related types of summation)? If we have a function we wish to sum $f(n)$, but $$ S^0[f] = \sum_{n=1}^\infty f(n) $$ diverges, ...
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1answer
79 views

(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 ...
3
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2answers
655 views

Limit of the Cesaro sum of the product of 0-1 sequences.

Assume that $a_n$ and $b_n$ are 0-1 sequences such that $$ \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N a_n = \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N b_n = p. $$ Let also $c_n$ an other 0-1 sequence. ...
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0answers
137 views

If a series of complex numbers $\sum_{n\in{\bf Z}_{\ge0}}c_{n}$ converges to $s$ then $\sum_{n\in{\bf Z}_{\ge0}}c_{n}$ is Cesàro summable to $s$

Prove that if a series of complex numbers $\displaystyle\sum_{n\in{\bf Z}_{\ge0}}c_{n}$ converges to $s$ then we have $\displaystyle\sum_{n\in{\bf Z}_{\ge0}}c_{n}$ is Cesàro summable to $s$ . My ...
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2answers
137 views

Show that $\sum c_n$ is not Cesaro summable

Consider the sequence $c_n=(-1)^{n-1}n$ Show that $\sum c_n$ is not Cesaro summable using the hint: "If $\sum c_n$ is Cesaro summable, then $\frac {c_n}{n}$ tends to $0$" the $N^{th}$ Cesaro sum of ...
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1answer
120 views

Convergence of a Cesaro sequence

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence such that $a_i\in[0,1]$ for every $i\in \mathbb{N}$, and suppose that $$\lim_{n \to \infty}\frac{1}{n} \sum_{i=1}^n a_i = p.$$ Does $$\frac{1}{n} \sum_{i=1}^...
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1answer
102 views

Follow-up question: Cesaro mean of cesaro mean of …

In this post, Cesaro mean of Cesaro means, benny asked about a bounded sequence for which the Cesàro mean diverges, but the Cesàro mean of the Cesàro mean converges (which still isn't answered ...
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2answers
368 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
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1answer
177 views

Is $(-1/2)^n$ Cesaro summable?

It is easy if $S_n=(-1)^n$; it is Cesaro summable to $0$. But I am unable to find if the sequence $S_n=(-1/2)^n$ is Cesaro summable or not.
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3answers
102 views

Show that $\displaystyle \lim_{n \to \infty}\sum_{i=1}^n \frac{((n+1)-i)a_i}{n^2} = \frac{a}{2} $

Assume $a_n \to a$. Then Show that $$\lim_{n \to \infty}\sum_{i=1}^n \frac{((n+1)-i)a_i}{n^2} = \frac{a}{2} $$ So as $a_n \to a$ we know $\forall \epsilon, \exists N, \forall n \ge N ,|a_n - a | &...
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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\...
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1answer
44 views

Property of Cesaro summable 0-1 sequences

Assume that $a_n$, $b_n$ and $c_n$ are 0-1 sequences such that $$ a=\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N a_n, \, c=\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N c_n, \, d=\lim_{N\to\infty} \frac{1}{...
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72 views

Dominated convergence theorem for vairance and cesaro mean of random variables

I was wondering the following problem: If $X_n$ is a sequence of independent random variables, $|X_n|\leq b, (b>0)$ and $X_n$ converges to $X$ almost surely. Using dominated convergence theorem, ...
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1answer
57 views

If $a_n$ and $a_n b_n$ are Cesàro summable 0-1 sequences, is $b_n$ Cesàro summable as well?

My question is related to this one: Is the product of a Cesaro summable sequence of $0$s and $1$s Cesaro summable? Let $(a_n)$ and $(b_n)$ be infinite sequences of zeros and ones. Assume that $\lim_{...
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6answers
225 views

Finding $\lim_{n\to\infty}\frac{1^p+3^p+…+(2n+1)^p}{n^{p+1}}$

I'm trying to solve the following problem: Find $$\lim_{n\to\infty}\frac{1^p+3^p+\ldots+(2n+1)^p}{n^{p+1}}$$ What I've got so far: My idea is to use Stolz-Cesaro theorem, which implies that: $$ \...
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0answers
537 views

Understanding part of a proof for Stolz-Cesaro Theorem

I'm trying to understand a step from a proof of the Stolz-Cesaro Theorem. Let ${\left\{ {{b_n}} \right\}_{n \in {\Bbb N}}}$ is a positively strictly increasing unbounded sequence. If ${\left\{ {{a_n}...
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3answers
102 views

proof verification of $\frac{1}{n}\sum \sin nx$ having a limit

I have to prove that $b_n=\frac{1}{n}\sum\limits_{i=1}^n \sin ix$ has a limit. I'm using the result of an already solved problem which implies the following: $$\sum_{i=1}^n \sin ix=\frac{\sin{\frac{...
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1answer
36 views

proving that if $lim_{n\to\infty}a_n=\infty$ then $lim_{n\to\infty}b_n=\infty$, where $b_n=\frac{1}{n}\sum_{i=1}^n{a_i}$.

I have to prove that if $lim_{n\to\infty}a_n=\infty$ then $lim_{n\to\infty}b_n=\infty$, where $b_n=\frac{1}{n}\sum_{i=1}^n{a_i}$. What I've got: Let $\epsilon > 0$. We know that $a_n\to \infty$, ...
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2answers
155 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_\...
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1answer
73 views

Proving the converse to the Cesaro theorem under weak assumptions

I have previously asked a question to prove the converse of the Cesaro theorem under the assumption that $u_{n+1}-u_n=o(\frac{1}{n})$ . This time I have to do it under the assumption that $u_{n+1}-u_n=...
2
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1answer
193 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} \...
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0answers
88 views

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 ...
2
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1answer
279 views

Limit of $n$-Cesaro summation as $n \to \infty$

I recently learned that a Cesaro summation extends the usual summation in the following way: Given a sequence $a_1, a_2, \ldots $ we construct the Cesaro sequence by defining $$\sigma_n = \frac{1}{n}\...
3
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1answer
77 views

Cesaro continiuity leads linearity

I need just a hint please. It seems that I have to prove that $f(x)=mx$ in which $m\in \mathbb{R}.$ But I couldn't handle it. Problem: We say that a sequence $x_{n}\; , n = 1, 2,\cdots ,$ ...
4
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4answers
328 views

Example of $(b_n)$ such that $\lim_{n\to\infty} {\frac1n}\sum_{i=1}^{n-1}b_i$ does not exists and $0\le b_n\le 1$

Find a ${{b_n}}$ $n\in\Bbb N$ and $0\le b_n\le 1$ such as the limit $$\lim_{n\to\infty} {\frac1n}\sum_{i=1}^{n-1}b_i$$ does not exist. I don't know how to deal with this problem, it seems to me that ...
4
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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 + \...
2
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0answers
85 views

multidimensional pointwise convergence of manipulated Fourier series - reference request

For a continuous complex-valued continuous function $f$ on the unit circle $\mathbb{T}$, we have that $f\ast K_n$ converges uniformly to $f$, where $K_n$ are the Fejér kernels defined by taking Césaro ...
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1answer
208 views

Prove that $a_n:=\sin(\log(n))$ isn't Cesaro summable

The Cesaro limit is defined to be $$\lim\limits_{N\to\infty}\frac{\sum\limits_{n=1}^Na_n}{N}.$$ On his latest blog post, Terence Tao mentions that this sequence isn't Cesaro summable. How does one ...
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1answer
78 views

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, ...
2
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1answer
289 views

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 ...
1
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1answer
87 views

Prove Cesàro mean using the weak law of large numbers

Is it possible to prove that the Cesàro mean of a converging sequence is the limit of the sequence through probabilities using the weak law of large numbers ? Has it ever been done ?
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0answers
294 views

Proofs with absolutely summable sequences

(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., ...