Questions tagged [cesaro-summable]

For questions about Cesàro summation and Cesàro summable sequences.

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Confusion on Baby Rudin Chapter 3 Exercise 14 (e)

The question posed is as follows: For $\{s_n\}$ a sequence of complex numbers, $\sigma_n = \frac{s_0 + s_1 + ... + s_n}{n+1}$, $a_n = s_n - s_{n-1}$, $| n a_n | \leq M < \infty$, $ \forall n \in \...
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Let $a_n$ and $x_n,y_n\ge0$ be sequences such that $(x_na_n)$is Cesaro summable, $mx_n\le y_n\le Mx_n$ for some $m,M>0$ , $|x_na_n|\le1$ and $x_n\to0$

Let $(a_n),(x_n)$ and $(y_n)$ be sequences of real numbers with $x_n,y_n\ge0$, $mx_n\le y_n\le M x_n$ for some $m,M>0$, $|x_n a_n|\le1$, $x_n\to0$ and $\lim\limits_{N\to\infty}\frac{1}{N}\sum\...
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Convergence speed of Cesaro mean

Consider a sequence $(a_n)$ satisfying $\lim_{n\to\infty} a_n = a$. Let $b_n = \frac{1}{n} \sum_{i=1}^n a_i$. I have already known that $\lim_{n\to\infty} b_n = a$. I am wondering is there any ...
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Is the average of two convergent series equal to the Cesàro sum of the alternating series?

If we have $(a_n)$ and $(b_n)$ such that $\sum a_n$ converges and $\sum b_n$ converges, I know that we do not necessarily have that $\sum c_n$ (where $(c_n)_n=a_0,b_0,a_1,b_1,\dots$ converges. But is ...
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Convergence of Power Series to Its Cesaro Sum sentence in Fourier proof

Well, I am learning about Fourier sum, and I encountered Cesaro sum in the proof of convergent uniformly of Fourier sum, I know that Fejér sentence says that: $\|f(x) - \sigma_n(f))\| < \epsilon .$ ...
segev ezra's user avatar
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Let $c_n\ge0$ be a sequence. Sufficient conditions of $(c_n)$ such that $\lim 1/n\sum\limits_{k=1}^n c_k z^k=0$ for $|z|=1,z\ne 1$

Let $(c_n)$ be a sequence of non-negative reals which is bounded below and above i.e. $m\le c_n\le M$ for some $m,M>0$. But this is not enough to say about the limit of $\frac{1}{n}\sum\limits_{k=1}...
Nafula12's user avatar
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What is the Cesaro sum of $(-1+1-1+1-1\ldots )$?

I have recently familiarized myself with the peculiar result of: $1-1+1-1+1\ldots=\frac{1}{2}$ Following this enlightment I was now interested in finding out whether the following infinite series has ...
iluvmath's user avatar
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The series 1-3+1-3+1-3... is (c,1) summable?

To prove the series $1-3+1-3+1-3+1-3+... $ is (c,1) summable. A series $\sum_{n=1}^\infty a_n $ is said to be (c,1) summable if the sequence of partial sums $s_n$ is (c,1) summable. A sequence ${s_n}$ ...
Lakshmi Priya's user avatar
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The sequence {1,0,0,1,0,0,1,0,0,...} is (C,1) summable

To prove that the sequence 1,0,0,1,0,0,1,0,0,... is (C,1) summable: [A sequence $\{s_n \}$ is (C,1) summable to $L$ if the sequence $\{\sigma_n \}$ converges to $L$ where $$\sigma_n= \frac{s_1+s_2+......
Lakshmi Priya's user avatar
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Find $\lim_{n \to +\infty }\frac{e^{u_{1}}+e^{\frac{u_{2}}{2}}+e^{\frac {u_{3}}{3}}+...+e^{\frac {u_{n}}{n}}-n}{\ln(n+1)}$ when $u_{n}$ converges to u

We have a first premilinary question that is: let $u_{n}$ be a sequence that converges to $u$, show that $$\lim_{n \to +\infty} \frac {e^{\frac {u_{n}} {n}} - 1} {\ln(1 + 1/n)} = u$$ this is easy with ...
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Integrand and Cesaro average: Extreme Values, Regular Variation and Point Processes

I'm trying to understand a line in a proof. We have positive $f$ with its derivative $f'(u) \to 0$ as $u \to \infty$. We want to prove $\lim_{t \to \infty} \frac{f'(t)}{t} = 0$. The proof says that as ...
Phil's user avatar
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Prove If $(x_n)$ is a convergent sequence, then the sequence $y_n = \frac{x_1 + \ \cdots \ + x_n}{n}$ converges to the same limit. (Cesaro means)

This was a homework question in a first course in real analysis that I had taken as an undergraduate. The question was exercise 2.3.11 from Stephen Abbott's "Understanding Analysis" 2nd ...
mebenot's user avatar
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Application of Cesaro lemma

I would like to show that the sequence of random variables $\frac{S_n}{n}$ defined below converges to zero. The proof requires the application of Cesaro lemma and I don't understand how. First, let ...
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Cesàro mean of $P(X_k = \pm k)=\frac{1}{2k\log(k+1)}$, $P(X_k = 0)=1-\frac{1}{k\log(k+1)}$

For the sequence $(X_n)_{n\geq 1}$ of independent random variables as described in the title, I would like to know if the sequence $Y_n:=\frac{1}{n}\sum_{k=1}^{n}X_k$ converges in probability (or even ...
Math Enjoyer's user avatar
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Convergence of a double Cesaro mean

Am asked to show that $$\frac{1}{n}\sum_{i=1}^n\sum_{j=1}^n \rho^{|i-j|}\to \frac{1+\rho}{1-\rho} \quad \text{as} \quad n\to\infty$$ where $\rho$ is a number satisfying $|\rho|<1$. My attempt: ...
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Explaining a relation used in a proof of "Cesaro summable implies Abel summable"

In this post: Cesaro summable implies Abel summable, in the proof that Cesaro summable implies Abel summable, the following relation is used: $$\sum_{n=0}^\infty s^1_n x^n = (1-x)^{-1} \sum_{n=0}^\...
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Neccesary condition for an integral to be finite

Suppose that $(X,\Sigma,\mu)$ is a measurable space and $f$ a non-negative measurable function such that $$ \int_{X}{f}< +\infty $$ I want to prove that $\sum_{n=0}^{\infty}{2^n \mu( \left\{{x:f(x) ...
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The closure of certain subspace of $\ell_\infty$

What is the closure of the following subspace $A$ of $\ell_\infty$ with the standard sup norm of $\ell_\infty$: $$A=\{(a_n)\in \ell_\infty\mid (A_n)=a_1+a_2+\ldots+a_n\; \text{belongs to} \;\ell_\...
Ali Taghavi's user avatar
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Applications of Cesaro's lemma to probability theory

I am currently trying to collect applications of Cesaro's lemma to various fields. I suspect that there are (elementary) applications of this lemma to probability theory but couldn't find any. ...
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Cesaro summation of the inverse Fourier transform.

Let $f\in \mathcal{L}(\mathbb{R}^1).$ Prove that $$f(x)=\frac{1}{\sqrt{2\pi}}\lim_{T\to+\infty}\frac{1}{T}\int_{0}^{T}\int_{-t}^{t}e^{ixy}\hat fdy\,dt$$ for almost all x including points of continuity ...
ламер's user avatar
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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 $(u_n)_{n \in \mathbb{N}}$, $$\...
Pascal's user avatar
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Weakly null sequence in Schreier space

Define the Schreier space $X:=\overline{c_{00}}^{\Vert\cdot\Vert}$ where $$\Vert x\Vert = \sup\bigg\{\sum_{i=1}^k|x_{n_i}|:k\le n_1<n_2\cdots <n_k\bigg\}.$$ Show that there exists a weakly null ...
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How to prove the Divergence of cesaro mean of harmonic series [duplicate]

How do you prove that the Cesàro mean of the harmonic series diverges. I know that the harmonic serie $$\sum_{k=1}^{\infty} \frac{1}{k}$$ diverges, However, I'm having trouble proving that it also ...
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Is the series $\sum^{\infty}_{n=1}(-1)^{n-1}\frac{-2n-1}{n^2+n}$ Cesaro summable?

The sequence of terms converges to $0$, as the the sequence is convergent in the standard sense. One can quickly prove this as $\left|(-1)^{n-1}\frac{-2n-1}{n^2+n}\right|=\left|\frac{2n+1}{n^2+n} \...
jeb2's user avatar
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Determining whether the series $\sum^\infty_{n=1}(-1)^{n}\sqrt{n}$ is is Cesaro summable

In otherwords, whether the limit $\lim_{n\to\infty}\frac{1}{n}\sum^{n}_{k=1}S_k$ exists, where $S_k$ is the $k$th partial sum. I'm not quite sure where to start with this one, especially since the ...
jeb2's user avatar
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3 answers
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Cesàro sum of the alternating harmonic series

Consider $$ \sum^{\infty}_{k=1}(-1)^{k-1}\frac{1}{k} \tag{a} $$ where the Cesàro sum is $\lim_{k\to\infty}\frac{\sum^k_{j}s_j}{k}$ where $s_k$ is the $k$th partial sum. Is $(a)$ Cesàaro convergent? We ...
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Can someone detail why the following limit is true?

We have some $m \times m$ matrix $\Theta_i$ such that $\underset{i \rightarrow \infty}{\lim} \Theta_i = \bar{\Theta}$ where it is understood that all entries are finite. Furthermore, we have $\Sigma$, ...
Stéphane's user avatar
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Does every Cesaro summable sequence have bounded partial sums, is the sums are all positive.

If $a_n \in \mathbb{R}$, $A_N = \sum_{n=1}^N a_n$, all partial sum so $A_N$ are positive, but their sequence oscillates and doesn't not converge, It probably down to matter but all $a_n$ are not ...
Shree's user avatar
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Cesaro means $\frac{1}{N} \sum_1^{N} a_n$ of a positive sequence converging to zero while the sequence itself does not [duplicate]

In this post: Cesaro means of a positive sequence, OP is referring to a "famous example" of a sequence $a_n \geq 0$ whose Cesaro means $\frac{1}{N} \sum_1^{N} a_n$ converge to zero while the ...
undefined's user avatar
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2 answers
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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....
Mobius.Drip's user avatar
2 votes
2 answers
341 views

Convergence of an increasing sequence given convergence of the Cesaro sum

I’ve been playing with Cesaro summation and I’m now stuck on a problem. Given an increasing sequence $(u_n)_{n\in\mathbb{N}}$ of real numbers, define it’s Cesaro sum as $C_n = \frac{1}{n} \sum\limits_{...
t_kln's user avatar
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Equivalence on Cesàro convergence

Let $(x_n)$ be a bounded real sequence. Then $$ \lim_{n\to \infty}\frac{1}{n}\sum_{i=1}^n x_i=0 \Longleftrightarrow \lim_{n\to \infty}\frac{1}{2^n}\sum_{i=2^n+1}^{2^{n+1}} x_i=0. $$ Do you have a ...
Paolo Leonetti's user avatar
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1 answer
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Compute (if it exists) the one-sided limit of $\underset{x \rightarrow 1-}{\lim} \sum_{k=0}^{\infty} (-1)^k \ k \ x^k $

Compute (if it exists) the one-sided limit of $$\underset{x \rightarrow 1-}{\lim} \sum_{k=0}^{\infty} (-1)^k \ k \ x^k $$ I'm finding the question really confusing, especially the one-sided limit of $...
user9750060's user avatar
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1 answer
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What is the value of a sum similar to Basel problem but with fibonacci coefficient [closed]

The title is pretty self-explanatory. What is the value of $\sum_{n=1}^{\infty}\frac{f_n}{n^2}$
Lettever's user avatar
3 votes
3 answers
320 views

Cesàro sum of $1+ 0 - 1 + 1 + 0 - 1 + \dots$

I am trying to compute the Cesàro sum of $1+ 0 - 1 + 1 + 0 - 1 + \dots$. When I compute the Cesàro means, I get the following sequence $$\left(1, 1, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{4}{6}, ...
Vicky's user avatar
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Meaning of uniformly Cesàro summable

There is a theorem like: The Fourier series of a continuous function $f(x)$ defined on $[-\pi,\pi]$ ...
Esha's user avatar
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2 answers
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Is $1-2+3-4+…$ $2$-Cesàro summable to $1/4$?

According to the video https://www.youtube.com/watch?v=jcKRGpMiVTw, the series $s_n=\sum_{k=1}^n (-1)^{k+1} k$ is $2$-Cesàro summable. Because of the not-quite rigorous identity $$ s_\infty = \frac{1}{...
Rodrigo's user avatar
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6 votes
1 answer
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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 ...
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Prove that $\lim_{\lambda\rightarrow\infty} \frac{1}{\lambda}\int_0^\lambda\int_0^xf(y)\,dy\,dx = \int_0^\infty f(x)\,dx$ [closed]

Let $f:[0,\infty)$ be Lebesgue-integrable, then prove that $$\lim_{\lambda\rightarrow\infty} \frac{1}{\lambda}\int_0^\lambda\int_0^xf(y)\,dy\,dx = \int_0^\infty f(x)\,dx$$ This is also known as Cesàro ...
user801496's user avatar
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1 answer
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If all subsequences $\{x_{n_i}\}$ of $\{x_n\}$ $ \lim_k\frac1k\sum_{i=1}^{k} {x_{n_i}}= y $ then $\lim_n x_n= y$

Let $\{x_n\}_n$ be a real sequence and $y\in\mathbb{R}$ such that for all subsequences $\{x_{n_i}\}$ of $\{x_n\}$ we have $$ \lim_k\frac{1}{k}\sum_{i=1}^{k} {x_{n_i}}= y $$ My problem: Why $\lim_n ...
Karim KHAN's user avatar
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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}...
Bob's user avatar
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Generalized Cesàro summability of $(-1)^nn^p$

A sequence $\{a_n\}_{n\geq 0}$ is said to be $(C,\alpha)$-summable if $\lim_{n\to\infty} S^\alpha_n$ exists, where $$ S^n_\alpha = \sum_{k=0}^n \frac{ {n \choose k} }{ {n+\alpha \choose k} } a_k. $$ ...
Kenneth Ng's user avatar
4 votes
3 answers
187 views

Proof that $\sum_{n=0}^{\infty}(-1)^{n} = \frac{1}{2}$. Is there any error?

So, I proved that: $$\int f(\ln x)\ dx = x \sum_{n=0}^{\infty}(-1)^{n} f^{(n)}(\ln x) \ \ \ +\ \ C$$ where $f^{(n)}$ is the nth derivative of $f$. if we let $f(x) = e^{x}$ then $f^{(n)}(x) = e^x$ as ...
Eduardo Magalhães's user avatar
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0 answers
205 views

Cesàro Mean of Convergent Subsequence Converges?

Let $X$ be a compact and convex subset of $\mathbb R^n$ and let ${(x_{t})}_{t}$ be a sequence in $X$ such that $||x_{t+1}-x_{t}||_{\infty}\leq \frac{1}{t+1}$ for all $t=0,1,...$. Let ${(x_{t_n})}_{...
user_newbie10's user avatar
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1 answer
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Measure of Cesaro summable sequences

Loosely, I would like to know "how many" binary sequences are non-convergent in the sense of their average. Let $\{0,1\}^{\mathbb N}$ denote the set of all binary sequences. This set is homeomorphic ...
user_newbie10's user avatar
4 votes
1 answer
100 views

Does the Komlós theorem hold in infinite measure spaces?

I read an article, and they use a certain theorem, called Komlós theorem, which says: Let $(E,\mathcal {A}, \mu ) $ be a finite measure space and $ (f_n)_{n\geq 1} \subset \mathcal {L}_{\mathbb {R}...
Wer Wer's user avatar
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Unbounded series with finite Cesàro mean

Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of positive real numbers with finite Cesàro mean i.e., $$\lim\limits_{n\to\infty}\tfrac{1}{n}\sum_{i=1}^{n}a_i < \infty.$$ Prove or disprove $$ \lim\...
Shubhamoy Nandan's user avatar
1 vote
1 answer
188 views

Convergence of Cesaro sums on $L^p$

Let $K_N=\frac{1}{N}\sum_{n=0}^{N-1}D_n(x)$ be the Fejer kernel and let $\sigma_N(f)=\frac{1}{N}\sum_{n=0}^{N-1}S_N(f)$ where $S_n(f)=\frac{1}{\pi}\int_{-\pi}^{\pi}f(\tau)D_n(t-\tau)d\tau=f*D_n.$ With ...
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Proof of the integral representation of the arithmetic mean of the partial sums of Fourier series

The definition and the formula in question: The equations he refers to: ..................................................................................................................................
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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)\...
HumbleStudent's user avatar