Questions tagged [cesaro-summable]

For questions about Cesàro summation and Cesàro summable sequences.

Filter by
Sorted by
Tagged with
2
votes
1answer
380 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 ...
2
votes
0answers
160 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 ...
2
votes
0answers
87 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 ...
2
votes
1answer
292 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 ...
2
votes
0answers
233 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 ...
2
votes
1answer
76 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 ...
1
vote
4answers
341 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 ...
1
vote
1answer
210 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 ...
1
vote
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 $(s_n)_n$ ...
1
vote
1answer
275 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
3answers
102 views

proof verification of $\frac{1}{n}\sum \sin nx$ having a limit

I have to prove that $b_n=\frac{1}{n}\sum\limits_{i=1}^n \sin ix$ has a limit. I'm using the result of an already solved problem which implies the following: $$\sum_{i=1}^n \sin ix=\frac{\sin{\frac{...
1
vote
1answer
106 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 $n$ ...
1
vote
1answer
34 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}...
1
vote
1answer
104 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 ...
1
vote
1answer
41 views

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}{...
1
vote
1answer
94 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 ...
1
vote
2answers
140 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 ...
1
vote
1answer
487 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 ...
1
vote
1answer
97 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?
1
vote
1answer
247 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 ...
1
vote
1answer
62 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
27 views

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)\...
1
vote
0answers
88 views

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 ...
1
vote
1answer
89 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 ?
1
vote
1answer
99 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 ...
1
vote
2answers
222 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 ...
1
vote
1answer
178 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 ...
1
vote
0answers
48 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?
0
votes
2answers
314 views

Suppose that a sequence is Cesaro summable. Prove…

Suppose that a sequence $a_{n}$ is Cesaro summable. Prove that $$\lim_{n \to \infty }\frac{a_{n}}{n}=0$$
0
votes
1answer
83 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 $n$) $$ \frac{a_n - a_{n+1}}{a_n} \approx \frac{a_n}{na_n} = \frac{1}{n} \text{ }?$$ ...
0
votes
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 ...
0
votes
1answer
57 views

If $a_n$ and $a_n b_n$ are Cesàro summable 0-1 sequences, is $b_n$ Cesàro summable as well?

My question is related to this one: Is the product of a Cesaro summable sequence of $0$s and $1$s Cesaro summable? Let $(a_n)$ and $(b_n)$ be infinite sequences of zeros and ones. Assume that $\lim_{...
0
votes
1answer
287 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 ...
0
votes
1answer
36 views

proving that if $lim_{n\to\infty}a_n=\infty$ then $lim_{n\to\infty}b_n=\infty$, where $b_n=\frac{1}{n}\sum_{i=1}^n{a_i}$.

I have to prove that if $lim_{n\to\infty}a_n=\infty$ then $lim_{n\to\infty}b_n=\infty$, where $b_n=\frac{1}{n}\sum_{i=1}^n{a_i}$. What I've got: Let $\epsilon > 0$. We know that $a_n\to \infty$, ...
0
votes
1answer
100 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\...
0
votes
1answer
71 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 (...
0
votes
1answer
46 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 ...
0
votes
0answers
11 views

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: ..................................................................................................................................
0
votes
0answers
15 views

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 ...
0
votes
0answers
26 views

Geometric meaning of the Césaro limit of Geometric sequence on the Torus.

As a motivattion for an introdutory notion in our Ergodic lecuture, we were asked to give a Geometric meaning to the following limit. Let $\lambda\in \Bbb T$ $$\lim_{n\to \infty}\frac1n\sum_{k=0}^{...
0
votes
1answer
86 views

Hardy's power series. Cesàro convergence

Doing my analysis homework i have come across the following power series known as Hardy's power series $$\sum\limits_{k=0}^{\infty}a_kx^k=\sum\limits_{k=0}^{\infty}(-1)^kx^{2^k}\mbox{ for x}\in[0,1],$$...
0
votes
0answers
50 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, ...
0
votes
1answer
108 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 ...
0
votes
1answer
44 views

Property of Cesaro summable 0-1 sequences

Assume that $a_n$, $b_n$ and $c_n$ are 0-1 sequences such that $$ a=\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N a_n, \, c=\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N c_n, \, d=\lim_{N\to\infty} \frac{1}{...
0
votes
0answers
76 views

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, ...
0
votes
1answer
74 views

Proving the converse to the Cesaro theorem under weak assumptions

I have previously asked a question to prove the converse of the Cesaro theorem under the assumption that $u_{n+1}-u_n=o(\frac{1}{n})$ . This time I have to do it under the assumption that $u_{n+1}-u_n=...
0
votes
0answers
297 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., ...
0
votes
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 $a$?
0
votes
0answers
110 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{...
0
votes
1answer
215 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 ...