Questions tagged [cesaro-summable]

For questions about Cesàro summation and Cesàro summable sequences.

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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
386 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
96 views

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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4answers
336 views

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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108 views

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
353 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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1answer
87 views

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
383 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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345 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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1answer
132 views

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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1answer
37 views

$\liminf_{N\to\infty}\frac{1}{N}\sum_{n=1}^Na_nb_n\ge\liminf_{N\to\infty}\frac{1}{N}\sum_{n=1}^Na_n\liminf_{N\to\infty}\frac{1}{N}\sum_{n=1}^Nb_n$?

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
109 views

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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1answer
134 views

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
350 views

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
105 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
374 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
531 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{...
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1answer
107 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
702 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 ...
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1answer
79 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
2k 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
359 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
436 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
277 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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110 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}^...
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2answers
593 views

Is the Cesàro summation of a sequence divergent to infinity divergent?

More specifically the question is: If I have a sequence $(u_n)_{n\in\mathbb{N}} \subset \mathbb{R}$ that diverges to infinity. Then it's Cesàro summation sequence $(s_n)_{n\in\mathbb{N}}=(\frac{1}{n+1}...
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1answer
766 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 ...
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1answer
600 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 ...
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1answer
291 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
358 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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76 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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3answers
443 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
274 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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137 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
99 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 $$ \frac{1}{2}+ \sum_{k=1}^{\infty}\cos(kx) = 0, \quad (C,1). $$ Solution: So I know where ...
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1answer
269 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
86 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
76 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
102 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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4answers
584 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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53 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
193 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
382 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
124 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
673 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
108 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
323 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
279 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 ...
3
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0answers
281 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
69 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 ...