Questions tagged [cesaro-summable]
For questions about Cesàro summation and Cesàro summable sequences.
102
questions
2
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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
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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
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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
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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
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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}
$$
how to prove that :
$\...
1
vote
1answer
27 views
If I know the limit of Cesaro averages , then can I know limits of uniform Cesaro averages. Details Below
Say that I have a sequence $\{a_{n}\}_{n\in\mathbb{N}}$ $\subset$ [0,1] such that lim$_{N\rightarrow\infty}$ $\frac{1}{N}$ $\sum_{n=1}^{N}$ a$_{n}$ = a.
Can I somehow get the value of lim$_{(N-M)\...
1
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0answers
88 views
Summability Question
Today I was reading Cesaro Summability and Abel summability. I found that there exists a series which is Cesaro summable but do not converge in conventional way (the usual way...). Again there exists ...
1
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1answer
89 views
Prove Cesàro mean using the weak law of large numbers
Is it possible to prove that the Cesàro mean of a converging sequence is the limit of the sequence through probabilities using the weak law of large numbers ? Has it ever been done ?
1
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1answer
99 views
Cesaro summability of this sequence
Consider the sequence $(0,0,1,1,1,1,0,0,0,0,0,0,0,0,1,1..)$, i.e the sequence of 2 zeros, followed by $2^2$ ones, followed by $2^3$ zeros, $2^4$ ones and so on. I know that the sequence of Cesaro ...
1
vote
2answers
222 views
Limit of partial sums: $\lim_{n\to\infty} \frac1n\sum_{k=1}^n f(k)=0$ if $\lim_{k\rightarrow\infty}f(k)=0$ [duplicate]
I want to argue that
$$\lim_{n\rightarrow\infty} \frac{1}{n}\sum_{k=1}^n f(k)=0~~~~~~~ {\rm if}~~~~~ \lim_{k\rightarrow\infty}f(k)=0.$$
This identity does not seem to hold always, but seems to hold ...
1
vote
1answer
178 views
Convergence of infinite products
I wonder, parallel to the theory of summability of infinite series is there a theory for infinite products?
Is there any generalized convergence method (such as Cesaro and Abel summability) for the ...
1
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0answers
48 views
nonlinear sequence to sequence transformations
i know matrix methods such as Cesaro,Holder,Riesz are regular linear sequence transformations.
i wonder if there is any regular nonlinear sequence transformation?
0
votes
2answers
314 views
Suppose that a sequence is Cesaro summable. Prove…
Suppose that a sequence $a_{n}$ is Cesaro summable. Prove that
$$\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{ }?$$
...
0
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3answers
87 views
Show that the sum equals $0$ according to Cesàro
I'm stuck at some problems in my Fourier Analys course, maybe you got a clue.
If $x \neq n \cdot 2 \pi$, then
$$
\frac{1}{2}+ \sum_{k=1}^{\infty}\cos(kx) = 0 \tag{C,1}
$$
Solution:
So I know where ...
0
votes
1answer
57 views
If $a_n$ and $a_n b_n$ are Cesàro summable 0-1 sequences, is $b_n$ Cesàro summable as well?
My question is related to this one: Is the product of a Cesaro summable sequence of $0$s and $1$s Cesaro summable?
Let $(a_n)$ and $(b_n)$ be infinite sequences of zeros and ones.
Assume that
$\lim_{...
0
votes
1answer
287 views
Sum of the multiplicative function $\lambda_2$
Do we know any precise evaluation of the sum:
$$\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
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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 ...
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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
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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 $a$?
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 ...
