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Questions tagged [cesaro-summable]

For questions about Cesàro summation and Cesàro summable sequences.

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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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62 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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86 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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55 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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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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38 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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(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 ...
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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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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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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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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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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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122 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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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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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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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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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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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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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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2answers
374 views

General Cesaro summation with weight

Assume that $a_n\to \ell $ is a converging sequence of complex numbers and $\{\lambda_n\}$ is a sequence of positifs real numbers such that $\sum\limits_{k=0}^{\infty}\lambda_k = \infty$ Then, show ...
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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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3answers
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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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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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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=...
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1answer
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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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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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1answer
204 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}\...
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1answer
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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 $x_{n}\; , n = 1, 2,\cdots ,$ ...
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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 ...
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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
169 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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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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1answer
170 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 ...
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218 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., ...
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Question about Cesàro summation

Consider $$S_n = \sum_{i=0}^n a_i$$ and its Cesàro sums, defined as $$ C = \lim_{n \to \infty} \frac1n\sum_{k=0}^n S_k$$ Is it always true that $$ C = \lim_{n \to \infty} \frac1{L(n)}\sum_{k= n - L(n)}...
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$\liminf_{N\to\infty}\frac{1}{N}\sum_{n=1}^Na_nb_n\ge\liminf_{N\to\infty}\frac{1}{N}\sum_{n=1}^Na_n\liminf_{N\to\infty}\frac{1}{N}\sum_{n=1}^Nb_n$?

Suppose $a_n$ and $b_n$ are uniformly bounded sequences of non-negative numbers. Is it true that $$ \liminf_{N\to\infty} \frac{1}{N} \sum_{n=1}^N a_n b_n \ge \liminf_{N\to\infty} \frac{1}{N} \sum_{n=1}...
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1answer
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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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Applying Cesàro mean infinetely many times on 1+1+1+1… [closed]

It's well known that $\zeta(0)=\sum{\frac{1}{n^0}}=\sum{1}=-\frac{1}{2}$, so I know there is something wrong with extending the method Mathologer described here infinetely many times: Partial sums of ...
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1answer
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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: $\lim_{n \to \infty}\frac{1}{n}\sum_{k=0}^{n-1} ...
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1answer
69 views

Cesaro summability of this sequence

Consider the sequence $(0,0,1,1,1,1,0,0,0,0,0,0,0,0,1,1..)$, i.e the sequence of 2 zeros, followed by $2^2$ ones, followed by $2^3$ zeros, $2^4$ ones and so on. I know that the sequence of Cesaro ...
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1answer
190 views

Sum of the multiplicative function $\lambda_2$

Do we know any precise evaluation of the sum: $$\sum_{m \leqslant X} \lambda_2(m)$$ where $\lambda_2 = \mu \star \mu$ ? That is a first part to my question, which seems to me quite classical but I ...
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1answer
214 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$\hat{f}(...
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1answer
81 views

Cesaro mean of alternating sequences

How would you prove/show the cesaro mean $\lim_{n \to \infty}\left(\frac{1}{n}\sum_{k=1}^n a_k\right)$ of an aternating sequence such as: $$a_k = \begin{cases} 1 & k\equiv 0\mod 3 \\ 0 & k\not\...
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3answers
395 views

Can an unbounded sequence have a convergent cesaro mean?

I was wondering if an unbounded sequence may have a convergent cesaro mean ($\frac{1}{n}\sum_{k=1}^n a_n$). I was maybe thinking of $$a_n = (-n)^n$$ as a sequence having a convergent mean, but I might ...
4
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1answer
72 views

Prove that if $\sqrt[n]{\prod\limits_{i\leq n}a_i}$ converges to a finite limit then $a_n$ converges

Prove that if $$\lim_{n\to\infty} \sqrt[n]{\prod_{i\leq n}a_i} < \infty$$ then $\lim_{n\to\infty} a_n$ exists. Given that $\{a_i\}$ is bounded and positive. So I used Cesaro means to show that $\...
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2answers
1k 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 $(s_n)_n$ ...
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1answer
227 views

Does the converse to Kronecker's lemma hold?

Odds are that this question has been answered already and even that the argument is not too complicated, but here it goes: Assume that $(a_{k})_{k\in\mathbb{N}}$ is a sequence of real numbers and $(...
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1answer
253 views

What useful properties does usual summation have, but alternatives do not? (Cesaro, etc)

Before I really ask my question, I want to give my train of reasoning. Suppose we have some method of summation (as I understand, assigning a number to a series) that satisfies some or all of ...
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2answers
153 views

Limit of partial sums: $\lim_{n\to\infty} \frac1n\sum_{k=1}^n f(k)=0$ if $\lim_{k\rightarrow\infty}f(k)=0$ [duplicate]

I want to argue that $$\lim_{n\rightarrow\infty} \frac{1}{n}\sum_{k=1}^n f(k)=0~~~~~~~ {\rm if}~~~~~ \lim_{k\rightarrow\infty}f(k)=0.$$ This identity does not seem to hold always, but seems to hold ...
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2answers
102 views

Is boundedness required in equivalence between $\frac1n\sum_{k=1}^na_k\to0$ and $\frac1n\sum_{k=1}^na_k^2\to0$?

Suppose $a_n$ is a sequence of non-negative real numbers. If $a_n$ are un-bounded, then I want to know if $\dfrac{1}{n}\sum_{k=1}^na_k\to0$ as $n\to\infty$ is equivalent to $\dfrac{1}{n}\sum_{k=1}^...