# Questions tagged [cesaro-summable]

For questions about Cesàro summation and Cesàro summable sequences.

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### 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 ...
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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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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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### Showing a positive Cesaro summable series is normally summable.

Question (Spivak, 23-13): Suppose that $a_n>0$ and $\{a_n\}$ is Cesaro summable. Suppose also that the sequence $\{na_n\}$ is bounded. Prove that the series $\sum_{n=1}^{\infty}a_n$ converges. My ...
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### 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, ...
### 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 ...