Questions tagged [cesaro-summable]

For questions about Cesàro summation and Cesàro summable sequences.

Filter by
Sorted by
Tagged with
2
votes
1answer
442 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
votes
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}...
1
vote
1answer
197 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 ...
5
votes
2answers
472 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
votes
2answers
161 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 - \...
3
votes
2answers
827 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 \...
0
votes
1answer
87 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 $n$) $$ \frac{a_n - a_{n+1}}{a_n} \approx \frac{a_n}{na_n} = \frac{1}{n} \text{ }?$$ ...
3
votes
2answers
427 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 \...
3
votes
2answers
299 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?
5
votes
2answers
597 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 ...
3
votes
2answers
96 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 ...
8
votes
4answers
481 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 ...
0
votes
1answer
269 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 ...
0
votes
1answer
48 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
vote
1answer
63 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
vote
1answer
297 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.
1
vote
0answers
50 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?
17
votes
1answer
16k 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 - ...
10
votes
2answers
2k 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 + \...
31
votes
3answers
11k 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}...

1 2
3