Questions tagged [cesaro-summable]

For questions about Cesàro summation and Cesàro summable sequences.

98 questions
24 views

73 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 ...
130 views

Banach limit for Cesaro summable sequences

I'm solving an exercise from Lax's Functional analysis. The section concerns generalized limits (more particularly, Banach limits), which are obtained by applying the Hahn-Banach theorem to the ...
77 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 $$\sum\limits_{k=0}^{\infty}a_kx^k=\sum\limits_{k=0}^{\infty}(-1)^kx^{2^k}\mbox{ for x}\in[0,1],$$...
85 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 $n$ ...
46 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, ...
77 views

(Non-) Convergence of $\frac{1}{n} \sum_{k=0}^{n - 1} \exp\left(2i \pi [\frac{3 + \sqrt{5}}{2}]^k\right)$ when $n \to +\infty$

Let be $$\forall n > 0, S_n = \dfrac{1}{n} \sum\limits_{k=0}^{n - 1} \exp(2i\pi u_k),\quad \forall k \geq 0, u_k = \left(\dfrac{3 + \sqrt{5}}{2}\right)^k$$ I would like to prove or disprove the ...
633 views

Limit of the Cesaro sum of the product of 0-1 sequences.

Assume that $a_n$ and $b_n$ are 0-1 sequences such that $$\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N a_n = \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N b_n = p.$$ Let also $c_n$ an other 0-1 sequence. ...
108 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 ...
120 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 ...
89 views

67 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, ...
56 views

61 views

87 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 ...
235 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 ...
284 views

1k 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 $(s_n)_n$ ...
Odds are that this question has been answered already and even that the argument is not too complicated, but here it goes: Assume that $(a_{k})_{k\in\mathbb{N}}$ is a sequence of real numbers and \$(...