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Questions tagged [cesaro-summable]

For questions about Cesàro summation and Cesàro summable sequences.

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2answers
188 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
105 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}^...
3
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2answers
387 views

Is the Cesàro Sumation of Series divergent to infinity divergent?

More specifically the question is: If I have a series $(u_n)_{n\in\mathbb{N}} \subset \mathbb{R}$ that diverges to infinity. Then it's cesaro sumation series $(s_n)_{n\in\mathbb{N}}=(\frac{1}{n+1}\...
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1answer
624 views

Calculating the Cesaro sum of $1-1+0+1-1+0+\dots$

I am having difficulty understanding how to find the Cesaro sum of the series: $1-1+0+1-1+0+\dots$ I know the sequence of partial sums will be: $1,0,0,1,0,0,1,0,0,1,0,0,\dots$ And hence the ...
11
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1answer
505 views

Can we show that $1+2+3+\dotsb=-\frac{1}{12}$ using only stability or linearity, not both, and without regularizing or specifying a summation method?

Regarding the proof by Tony Padilla and Ed Copeland that $1+2+3+\dotsb=-\frac{1}{12}$ popularized by a recent Numberphile video, most seem to agree that the proof is incorrect, or at least, is not ...
2
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1answer
223 views

Rate of convergence of Cesàro means

For a sequence $a_n = O(n^{-1/2})$ as $n\to\infty$, consider the corresponding Cesàro means $b_n = \frac{1}{n} \sum_{j=1}^n a_j$. Is it possible to derive the rate of convergence for the sequence $...
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2answers
299 views

Suppose that a sequence is Cesaro summable. Prove…

Suppose that a sequence $a_{n}$ is Cesaro summable. Prove that $$\lim_{n \to \infty }\frac{a_{n}}{n}=0$$
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1answer
70 views

Showing this piecewise series is not Cesaro summable

I have been asked to show that the series a_{n}= \begin{matrix} \frac{n+1}{2}&if&n&is&odd& & \\ -\frac{n}{2}&if&n&is&even & & \end{matrix} is not (...
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2answers
345 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 ...
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0answers
175 views

General Cesàro summability (and examples?)

According to Wikipedia, given a series $\sum a_n$ we can define a general Cesàro sum (C, $\alpha$) for $\alpha \in \Bbb R \setminus \Bbb N$ as $\lim_{n\to\infty}\dfrac {A^\alpha_n}{E^\alpha_n}$ where ...
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132 views

Cesaro and Tandori sequence spaces, representations and duality

Definitions. Fix $1\leq p\leq\infty$. Given a scalar sequence $a=(a_n)_{n=1}^\infty$, denote by $\tilde{a}=(\tilde{a}_n)_{n=1}^\infty$, where each $\tilde{a}_n=\sup_{k\geq n}|a_k|$. Now we define ...
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3answers
87 views

Show that the sum equals $0$ according to Cesàro

I'm stuck at some problems in my Fourier Analys course, maybe you got a clue. If $x \neq n \cdot 2 \pi$, then $$ \frac{1}{2}+ \sum_{k=1}^{\infty}\cos(kx) = 0 \tag{C,1} $$ Solution: So I know where ...
3
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1answer
226 views

Cesaro summation

Let's consider $\{a_{n} \} -$ a bounded sequence of real numbers. Is it true that $\frac{1}{n} \sum_{k=0}^{n-1}{|a_{k}|^{p}}$ (Cesaro sums) converges or diverges for all $p \geq 1$? (more presicely: ...
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1answer
72 views

Cesaro limit of analytic functions

Let $f_n$ be a uniformly bounded sequence of analytic functions on $\Omega\subset\mathbb C$. If $f_n(z)\to f(z)$ forall $z\in\Omega$, then by the Montel's theorem I know that the convergence is ...
2
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1answer
67 views

Let a sequence of real numbers $(s_n)$ is one-sided bounded. Is the Cesaro mean of the sequence $(s_n)$ also one-sided bounded?

Let $(s_n)$ be a sequence of real numbers and $(s_n)\geq -C$ for some $C\geq 0.$ I wonder if the sequence of Cesaro means of $(s_n)$ $$\sigma_n(s)=\frac{1}{n+1}\sum_{k=0}^{n}s_k$$ is also one-sided ...
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1answer
97 views

Cesáro summation, prove convergence [closed]

Prove that if $a_k≥0$ and $\Sigma \, a_k$ is $(C,1)-summable$ (Cesáro summable), then the series is convergent in the usual sense. (Assume the contrary – what does that entail for a positive series?) ...
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483 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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0answers
51 views

Prove that $\lim_{n\to\infty} a_n = a$ implies $\lim_{n\to\infty} \frac 1n\sum_{k=0}^n a_k = a$ [duplicate]

How can I prove that if a sequence $a_n$ has a limit $a$, then the arithmetic mean $$\frac {1}{n}\sum_{k=0}^n a_k$$ has the same limit $a$?
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1answer
152 views

Does Cesàro continuity imply continuity?

Following the known definition "a function is continuous at $x_0$ iff for any sequence $(u_n)_\Bbb{N}$ converging to $x_0$, the sequence $(f(u_n))_\Bbb{N}$ converges to $f(x_0)$", let's say that a ...
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4answers
319 views

If $\lim_{n\to\infty}a_{n}=l$, Then prove that $\lim_{n\to\infty}\frac{a_{1}+a_2+\cdot..+a_n}{n}=l$ [duplicate]

Given $a_n$ be a sequence and IF $\lim_{n\to\infty}a_{n}=l$, Then prove that $\lim_{n\to\infty}\frac{a_{1}+a_2+\cdot..+a_n}{n}=l$ I do not know how to do this. Can someone help me with this? Thanks ...
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0answers
104 views

Cesaro sum of a series [duplicate]

$\sum_{n=0}^{\infty}a_n$ diverges in the regular term but is Cesaro summable Prove $a_n/n\to 0$ when $n\to \infty$ We used the definition of the Cesaro sum and obtained: $\lim_{N\to \infty}\frac{...
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1answer
424 views

Harmonic Series and Its Divergence by Abel Sum and Cesaro Sum

Already I know that harmonic series, $$\sum_{k=1}^n\frac1k $$ is divergent series. And, it is also divergent by Abel Sum or Cesaro Sum. However, I do not know how to prove it is divergent by concept ...
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1answer
90 views

Is the Cesaro Operator normal?

The Cesàro operator $T:ℓ_p→ℓ_p$ is defined by $$(Tx)_k=(1/k)\sum_{j=1}^k x_j$$ where $x=(x_j)$. Is this operator normal?
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1answer
250 views

Divergent succession, but with convergent sum average.

An example of a sequence $a_n$ such that: $$a_n\rightarrow\pm\infty$$ but $$b_n=\frac{\sum_{k=1}^{n}a_k}{n}$$ converge.
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1answer
240 views

Cesàro means of divergent series

Does $\sum \limits_{n=2}^\infty n$ have a greater Cesàro mean than $\sum \limits_{n=1}^\infty n$? If not, then is there any other sense of "mean" in which the former's mean is greater than the ...
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0answers
270 views

Equivalent of Cesaro or Abel Summation for Limits

Functions such as sin(x) are not considered to have limits as x approaches infinity. Sequences such as Grandi's series of 1-1+1-1+1... are not considered to have sums classically but with expanded ...
3
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1answer
65 views

Summation approaches zero

if I have a sequence $A=$ $\frac{1}{n} \sum _{k=1}^n kC_k$ and $nC_n$ approaches $0$ as $n$ approaches infinity, then how can I show that $A$ goes to $0$? I just need it for my proof of the Tauber's ...
2
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1answer
355 views

Cesaro means of uniformly convergent sequence of functions also converges

Statement of the problem: Prove: If a sequence of complex functions $s_n$ on a set $X$ converges uniformly to a complex function $s$, then the sequence of Cesaro means $\sigma_N$ also converges ...
3
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1answer
1k views

Cesaro summable series

A series $\sum_{k=0}^{∞}a_k$ is said to be Cesaro summable to an $L\in R$ if and only if $\sigma_n = \sum_{k=0}^{n-1}(1 - \frac{k}{n})a_k$ converges to $L$ as $n$ → $∞$. Let $s_n = \sum_{k=0}^{n-1}...
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1answer
167 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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2answers
319 views

Cesáro sums and the actual limit

My textbook, as an aside, defines the Cesáro sum as follows: $$ \sigma_n= \frac{s_1+...+s_n}{n}= \frac{1}{n}\sum_{k=1}^ns_k, $$ where $$ s_n = \sum_{k=1}^na_k. $$ This method is used, I am told, to ...
3
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2answers
130 views

asymptotics of sum

I wanna find asymptotic of sum below $$\sum\limits_{k=1}^{[\sqrt{n}]}\frac{1}{k}(1 - \frac{1}{n})^k$$ assume I know asymptotic of this sum (I can be wrong): $$\sum\limits_{k=1}^{n}\frac{1}{k}(1 - \...
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2answers
657 views

Cesàro summable sequences

During some homeworks the following question came into my mind (it is not part of the homeworks): Let $(a_k)_{k \in \mathbb{N}}$ be a Cesàro summable sequence in $\mathbb{C}$ and let $a := \lim_{n \...
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1answer
80 views

$\frac{a_n - a_{n+1}}{a_n} \approx \frac{1}{n}$? (part of 2010 Putnam exam)

Given a non-negative sequence $a_n$, strictly decreasing and tending to zero, can we show that (for large $n$) $$ \frac{a_n - a_{n+1}}{a_n} \approx \frac{a_n}{na_n} = \frac{1}{n} \text{ }?$$ ...
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2answers
266 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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2answers
205 views

Cesàro summability and $\sum n \lvert a_n\rvert ^2 < \infty$ implies convergence

How can I prove that if $\sum_{n=1}^\infty a_n$ is Cesàro summable and if $\sum_{n=1}^\infty n |a_n|^2 < \infty$, then $\sum_{n=1}^\infty a_n$ converges?
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2answers
455 views

A result on sequences: $x_n\to x$ implies $\frac{x_1+\dots+x_n}n\to x$ without using Stolz-Cesaro [duplicate]

If $x_n \to x$, how might we prove $$\lim_{n \to \infty} \frac{\sum_{i=1}^{n} x_i}{n} = x$$ Of course, one has $\limsup x_n = \liminf x_n = x$, and thus, using the Stolz-Cesaro theorem: $$\liminf ...
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2answers
84 views

Show that for a sequence of real numbers $(a_n)_n$ $\lim_n a_n=0$ implies $\frac{1}{n}\sum_{i=0}^{n-1}\lvert a_i\rvert=0$

Let $(a_n)_{n\in\mathbb{N}}$ be q sequence of real numbers with $\lim_{n\to\infty}a_n=0$. Show that this implies $$ \lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}\lvert a_i\rvert=0. $$ This is my idea ...
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4answers
340 views

Does $\sum_{i=1}^\infty a_i/i < \infty$ imply that $a_i$ has Cesaro mean zero?

If $(a_i)_{i=1}^\infty$ is a sequence of positive real numbers such that: $$ \sum_{i=1}^\infty \frac{a_i}{i} < \infty. $$ Does this mean that the sequence $(a_i)_{i=1}^\infty$ has Cesaro mean ...
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1answer
202 views

Cesaro summability together with $\lim nu_n\to 0$ implies convergence

Assume the series $\sum u_n$ is Cesaro summable and $\lim_{n\to\infty} nu_n\to 0$. We want to see that the series is (Cauchy) convergent. Attempt: Let $s_n=\sum_{i=1}^n u_n$ denote the $n$-th partial ...
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1answer
45 views

If $r_n\to r$ and $s_n\to s$, then $(r \star s)_M/M \to rs$.

I was going to ask this question, but I think I figured it out, so I thought I'd post my answer: In this question of mine, a user's answer makes the following claim: Suppose $r_n$ and $s_n$ are ...
1
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1answer
60 views

problem with lim of a sequence

i don't know how to prove this lim..i thought i can use Cesaro-Stolz here but i can't..: $$ \lim_{n \to \infty }a_{n}= \infty \\ b_{n}= \frac{1}{n}\sum_{k=1}^{\infty }a_{k} $$ how to prove that : $\...
1
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1answer
265 views

the limit of infinite sums using stolz cesaro theorem

I need to find the limit as n goes to infinity of $$ {1\over\sqrt{n^2+1}}+{1\over\sqrt{n^2+2}}+\cdots+{1\over\sqrt{n^2+n+1}}. $$ I'm trying to figure out if you can use shtolz theorem.
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0answers
46 views

nonlinear sequence to sequence transformations

i know matrix methods such as Cesaro,Holder,Riesz are regular linear sequence transformations. i wonder if there is any regular nonlinear sequence transformation?
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1answer
13k views

Convergence of series implies convergence of Cesaro Mean. [duplicate]

Proof. Let $\sum_{k = 0}^N c_k \rightarrow s$, let $\sigma_N = (S_0 + \dots + S_{N-1})/N$ be the $Nth$ Cesaro sum where $S_K$ is the $Kth$ partial sum of the series. Then $s - \sigma_N \\= s - c_0 - ...
8
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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\...
4
votes
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 + \...
24
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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}...