# Questions tagged [cesaro-summable]

For questions about Cesàro summation and Cesàro summable sequences.

101 questions
Filter by
Sorted by
Tagged with
9k views

534 views

### 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 ...
1k views

508 views

302 views

290 views

274 views

### Equivalent of Cesaro or Abel Summation for Limits

Functions such as sin(x) are not considered to have limits as x approaches infinity. Sequences such as Grandi's series of 1-1+1-1+1... are not considered to have sums classically but with expanded ...
576 views

### General Cesaro summation with weight

Assume that $a_n\to \ell$ is a convergent sequence of complex numbers and $\{\lambda_n\}$ is a sequence of positive real numbers such that $\sum\limits_{k=0}^{\infty}\lambda_k = \infty$ Then, show ...
66 views

### $a_n=(-1)^{n-1}, \; s_n=\sum_{i=1}^{n}a_i$ then find $\lim_{n\to \infty}\frac{s_1+s_2+\dots s_n}{n}$

$a_n=(-1)^{n-1}, \; s_n=\sum_{i=1}^{n}a_i$ then find $\lim_{n\to > \infty}\frac{s_1+s_2+\dots s_n}{n}$ $$s_k=1,\; \text{if k is odd and } s_k=0 \text{ if k is even}$$ Cauchy's theorem for a ...
102 views

### Calculating the Cesaro sum of $1-1+0+1-1+0+\dots$
I am having difficulty understanding how to find the Cesaro sum of the series: $1-1+0+1-1+0+\dots$ I know the sequence of partial sums will be: $1,0,0,1,0,0,1,0,0,1,0,0,\dots$ And hence the ...