# Questions tagged [cesaro-summable]

For questions about Cesàro summation and Cesàro summable sequences.

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### Can we show that $1+2+3+\dotsb=-\frac{1}{12}$ using only stability or linearity, not both, and without regularizing or specifying a summation method?

Regarding the proof by Tony Padilla and Ed Copeland that $1+2+3+\dotsb=-\frac{1}{12}$ popularized by a recent Numberphile video, most seem to agree that the proof is incorrect, or at least, is not ...
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### Is $(-1/2)^n$ Cesaro summable?

It is easy if $S_n=(-1)^n$; it is Cesaro summable to $0$. But I am unable to find if the sequence $S_n=(-1/2)^n$ is Cesaro summable or not.
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### 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 ...
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### 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 ...
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 ...
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 \$(...