Questions tagged [cesaro-summable]

For questions about Cesàro summation and Cesàro summable sequences.

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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 ...
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1answer
43 views

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 $...
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1answer
21 views

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 $\sum_{n=1}^{\infty}\frac{f_n}{n^2}$
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3answers
96 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}, ...
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32 views

Meaning of uniformly Cesàro summable

There is a theorem like: The Fourier series of a continuous function $f(x)$ defined on $[-\pi,\pi]$ ...
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2answers
82 views

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}{...
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1answer
93 views

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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2answers
86 views

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 ...
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44 views

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 ...
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48 views

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}...
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35 views

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 $$ S^n_\alpha = \sum_{k=0}^n \frac{ {n \choose k} }{ {n+\alpha \choose k} } a_k. $$ ...
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13 views

Condition for the convergence of a variant of Cesaro mean

Let $\{z_n\}$ be a sequence of non-negative numbers. Then what can be the minimum condition such that, $\frac{1}{k^2}\sum_{i=1}^kz_i\rightarrow 0$ as $k\rightarrow\infty$. I got one, that is if $\{...
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3answers
163 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 ...
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46 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})}_{...
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20 views

$\sum_{k=1}^{\infty} k \mid x_k \mid^2$ and Cesàro summable implies summable.

I would like to show some result on Fourier series, but for that I have to show the intermediate step that for a sequence of complex numbers $\{x_k\}_{k \ge 1}$, if we have $\sum_{k=1}^{\infty} k \mid ...
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1answer
33 views

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 ...
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79 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}...
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54 views

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\...
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1answer
38 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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32 views

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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35 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
56 views

Relation between the sets of almost convergent, bounded statistical convergent and Cesaro convergent sequences

$\bullet$ A sequence $a=(a_n)$ is said to be Cesaro-summable or Cesaro-convergent to $l$ if the sequence $y=(y_n)$ defined by $y_n=\frac{a_1+a_2+a_3+\dots+a_n}{n}$, converges to $l$. $\bullet$ A ...
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89 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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45 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
107 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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231 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
97 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
121 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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1answer
82 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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85 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 ...
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696 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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2answers
199 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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220 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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1answer
142 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
137 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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1answer
210 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
125 views

Show that $ \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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1answer
55 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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120 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
63 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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1answer
42 views

Divergent sequence $(a_n)_{n\in\mathbb{N}}$ such that $(\frac{1}{n} \sum\limits_{j=1}^n a_j)_{n\in\mathbb{N}}$ converges?

I'm searching for a sequence that diverges as such $(a_n)_{n\in\mathbb{N}}$ but if inserted in $(\frac{1}{n} \sum\limits_{j=1}^n a_j)_{n\in\mathbb{N}}$ it converges.
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295 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
708 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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2answers
831 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
51 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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3answers
134 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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2answers
194 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
94 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=...
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1answer
272 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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96 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 ...