Questions tagged [cesaro-summable]

For questions about Cesàro summation and Cesàro summable sequences.

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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 ...
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Proving that if Sequence Converge then Series also Converge [closed]

Prove using Stolz–Cesàro theorem If $\lim\limits_{n→∞}a_n=α$ then, $\lim\limits_{n→∞}\frac{S_n}{n}=α$ I have tried using epsilon but I can not figure it out. Can someone please elaborate.
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8 votes
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237 views

$u_n$ 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 $(u_n)_{n \in \mathbb{N}}$, $$\...
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2 votes
1 answer
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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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What important details is missing in the proof of the following statement?

Let $(a_n)_{n\in \mathbb{N}}$ be a sequence of real numbers such that $a_n-a_{n-1}=o_{n\to+\infty}\left(\dfrac{1}{n}\right)$ which is Cesàro convergent. Then $(a_n)_{n\in \mathbb{N}}$ is convergent. ...
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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} \...
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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 ...
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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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2 answers
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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 $\Sigma$, ...
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1 vote
0 answers
60 views

Identity involving Cesaro limits

In Kingman (1978), we find the following "easily proved fact of elementary analysis": If for each $r\in[k]$ where $[k]=\{1,2,3,...k\}$ the sequence $a_r$ is bounded and has Cesaro sum $$\...
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What are some weak conditions on a bounded sequence $(a_n)$ that makes $z_n = \frac{a_1 + \dots +a_n}{n}$ converge as $n \to \infty\ ?$

Note, I have edited my question, due to realising that I mis-understood what is meant by a "Cesàro sum". What are the weakest conditions on a bounded sequence $(a_n)$ that makes the $z_n = \...
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1 answer
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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 ...
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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 ...
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6 votes
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....
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2 votes
2 answers
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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_{...
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3 votes
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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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1 vote
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 $...
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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 $\sum_{n=1}^{\infty}\frac{f_n}{n^2}$
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3 votes
3 answers
160 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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2 answers
141 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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3 votes
2 answers
187 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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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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2 answers
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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 ...
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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 ...
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0 answers
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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}...
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0 votes
0 answers
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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 $$ S^n_\alpha = \sum_{k=0}^n \frac{ {n \choose k} }{ {n+\alpha \choose k} } a_k. $$ ...
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4 votes
3 answers
176 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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1 vote
0 answers
132 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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0 votes
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 ...
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3 votes
1 answer
90 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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0 votes
1 answer
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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\...
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1 vote
1 answer
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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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0 answers
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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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1 vote
1 answer
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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)\...
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2 votes
1 answer
89 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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4 votes
1 answer
108 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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1 vote
1 answer
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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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  • 1,033
1 vote
1 answer
137 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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5 votes
0 answers
338 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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0 votes
1 answer
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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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1 vote
1 answer
174 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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1 vote
1 answer
110 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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6 votes
1 answer
113 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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  • 1,840
3 votes
2 answers
734 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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1 vote
2 answers
356 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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2 votes
0 answers
310 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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4 votes
1 answer
170 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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0 votes
1 answer
158 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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2 votes
1 answer
267 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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