# Questions tagged [cesaro-summable]

For questions about Cesàro summation and Cesàro summable sequences.

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### Equivalence on Cesàro convergence

Let $(x_n)$ be a bounded real sequence. Then $$\lim_{n\to \infty}\frac{1}{n}\sum_{i=1}^n x_i=0 \Longleftrightarrow \lim_{n\to \infty}\frac{1}{2^n}\sum_{i=2^n+1}^{2^{n+1}} x_i=0.$$ Do you have a ...
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### Proof that $\sum_{n=0}^{\infty}(-1)^{n} = \frac{1}{2}$. Is there any error?

So, I proved that: $$\int f(\ln x)\ dx = x \sum_{n=0}^{\infty}(-1)^{n} f^{(n)}(\ln x) \ \ \ +\ \ C$$ where $f^{(n)}$ is the nth derivative of $f$. if we let $f(x) = e^{x}$ then $f^{(n)}(x) = e^x$ as ...
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### Measure of Cesaro summable sequences

Loosely, I would like to know "how many" binary sequences are non-convergent in the sense of their average. Let $\{0,1\}^{\mathbb N}$ denote the set of all binary sequences. This set is homeomorphic ...
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### Divergent sequence $(a_n)_{n\in\mathbb{N}}$ such that $(\frac{1}{n} \sum\limits_{j=1}^n a_j)_{n\in\mathbb{N}}$ converges?

I'm searching for a sequence that diverges as such $(a_n)_{n\in\mathbb{N}}$ but if inserted in $(\frac{1}{n} \sum\limits_{j=1}^n a_j)_{n\in\mathbb{N}}$ it converges.
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How can we show that the bounded sequences which are Cesaro summable, i.e., the sequences such that the limit $$\lim\limits_{n\to\infty} \frac{x_1+\dots+x_n}n$$ exists, form a closed subset of $\ell_\... 1answer 94 views ### Proving the converse to the Cesaro theorem under weak assumptions I have previously asked a question to prove the converse of the Cesaro theorem under the assumption that$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=...
I'm trying to prove that, given $(u_n)_n \in \mathbb{C}^\mathbb{N}$ verifying $u_{n+1}-u_n =_{n} o(\frac{1}{n})$, the following holds:  \lim_{n\to\infty} \frac{u_1+...+u_n}{n} = a \in \mathbb{C} \...