All Questions
Tagged with sequences-and-series cesaro-summable
97
questions
2
votes
2
answers
180
views
Does the rate $\log n$ imply "almost harmonic"?
Let $\{c_k\}$ be a decreasing positive sequence such that $\sum_{k=1}^n c_k \sim \log n$. Does it say $c_k=O(1/k)?$
I have found a reference where it says that the converse is true. I tried to tackle ...
- 825
2
votes
1
answer
70
views
Differently defined Cesàro summability implies Abel summability
I am trying to solve the Exercise 10 of Section 5.2 of the book `Multiplicative Number Theory I. Classical Theory' by Montgomery & Vaughan. In the exercise, they define the Cesàro summability of ...
- 1,145
0
votes
1
answer
34
views
Sequence converges, then the mean also converges (withouht knowing the mean converges to the same value)
Maybe this question has been read a lot of timehere but I failed to find this kind of version. Let me explain.
Let $\{a_n\}$ be a convergent sequence, then $\left \{ \frac{S_n}{n} \right\}$ also ...
- 353
0
votes
3
answers
44
views
Let $a_n$ and $x_n,y_n\ge0$ be sequences such that $(x_na_n)$is Cesaro summable, $mx_n\le y_n\le Mx_n$ for some $m,M>0$ , $|x_na_n|\le1$ and $x_n\to0$
Let $(a_n),(x_n)$ and $(y_n)$ be sequences of real numbers with $x_n,y_n\ge0$, $mx_n\le y_n\le M x_n$ for some $m,M>0$, $|x_n a_n|\le1$, $x_n\to0$ and $\lim\limits_{N\to\infty}\frac{1}{N}\sum\...
- 1,130
0
votes
0
answers
41
views
Convergence speed of Cesaro mean
Consider a sequence $(a_n)$ satisfying $\lim_{n\to\infty} a_n = a$. Let $b_n = \frac{1}{n} \sum_{i=1}^n a_i$. I have already known that $\lim_{n\to\infty} b_n = a$. I am wondering is there any ...
- 43
0
votes
1
answer
52
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Let $c_n\ge0$ be a sequence. Sufficient conditions of $(c_n)$ such that $\lim 1/n\sum\limits_{k=1}^n c_k z^k=0$ for $|z|=1,z\ne 1$
Let $(c_n)$ be a sequence of non-negative reals which is bounded below and above i.e. $m\le c_n\le M$ for some $m,M>0$. But this is not enough to say about the limit of $\frac{1}{n}\sum\limits_{k=1}...
- 33
1
vote
1
answer
121
views
What is the Cesaro sum of $(-1+1-1+1-1\ldots )$?
I have recently familiarized myself with the peculiar result of:
$1-1+1-1+1\ldots=\frac{1}{2}$
Following this enlightment I was now interested in finding out whether the following infinite series has ...
- 49
1
vote
1
answer
149
views
The series 1-3+1-3+1-3... is (c,1) summable?
To prove the series $1-3+1-3+1-3+1-3+... $ is (c,1) summable.
A series $\sum_{n=1}^\infty a_n $ is said to be (c,1) summable if the sequence of partial sums $s_n$ is (c,1) summable.
A sequence ${s_n}$ ...
- 389
0
votes
0
answers
98
views
The sequence {1,0,0,1,0,0,1,0,0,...} is (C,1) summable
To prove that the sequence 1,0,0,1,0,0,1,0,0,... is (C,1) summable:
[A sequence $\{s_n \}$ is (C,1) summable to $L$ if the sequence $\{\sigma_n \}$ converges to $L$ where
$$\sigma_n= \frac{s_1+s_2+......
- 389
2
votes
1
answer
73
views
Find $\lim_{n \to +\infty }\frac{e^{u_{1}}+e^{\frac{u_{2}}{2}}+e^{\frac {u_{3}}{3}}+...+e^{\frac {u_{n}}{n}}-n}{\ln(n+1)}$ when $u_{n}$ converges to u
We have a first premilinary question that is:
let $u_{n}$ be a sequence that converges to $u$,
show that $$\lim_{n \to +\infty} \frac {e^{\frac {u_{n}} {n}} - 1} {\ln(1 + 1/n)} = u$$
this is easy with ...
0
votes
1
answer
32
views
Integrand and Cesaro average: Extreme Values, Regular Variation and Point Processes
I'm trying to understand a line in a proof.
We have positive $f$ with its derivative $f'(u) \to 0$ as $u \to \infty$.
We want to prove $\lim_{t \to \infty} \frac{f'(t)}{t} = 0$.
The proof says that as ...
- 1,710
3
votes
1
answer
1k
views
Prove If $(x_n)$ is a convergent sequence, then the sequence $y_n = \frac{x_1 + \ \cdots \ + x_n}{n}$ converges to the same limit. (Cesaro means)
This was a homework question in a first course in real analysis that I had taken as an undergraduate. The question was exercise 2.3.11 from Stephen Abbott's "Understanding Analysis" 2nd ...
- 66
2
votes
1
answer
64
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Convergence of a double Cesaro mean
Am asked to show that $$\frac{1}{n}\sum_{i=1}^n\sum_{j=1}^n \rho^{|i-j|}\to \frac{1+\rho}{1-\rho} \quad \text{as} \quad n\to\infty$$
where $\rho$ is a number satisfying $|\rho|<1$.
My attempt:
...
- 4,827
2
votes
2
answers
184
views
Neccesary condition for an integral to be finite
Suppose that $(X,\Sigma,\mu)$ is a measurable space and $f$ a non-negative measurable function such that
$$
\int_{X}{f}< +\infty
$$
I want to prove that $\sum_{n=0}^{\infty}{2^n \mu( \left\{{x:f(x) ...
- 135
3
votes
2
answers
161
views
The closure of certain subspace of $\ell_\infty$
What is the closure of the following subspace $A$ of $\ell_\infty$ with the standard sup norm of $\ell_\infty$:
$$A=\{(a_n)\in \ell_\infty\mid (A_n)=a_1+a_2+\ldots+a_n\; \text{belongs to} \;\ell_\...
- 868