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 $n$ goes to $\infty$. And the Abel sum of $(a_n)$ is $lim_{x\rightarrow 0}\Sigma_{n=0}^{\infty} a_n e^{-nx}$. (Assuming these limits exist, of course.) These sums equal the conventional sum of a series when the latter exists, but even when the latter does not exist these sums may still exist.
My question is, does there exist a metric on $\mathbb{R}$ such that for all sequences $(a_n)$, the sequence $(s_n)$ converges to the Cesaro sum of $(a_n)$ whenever the latter exists? Does there exist such a metric for the Abel sum?
Best Answer
No. Consider, for example, $a_n = (-1)^{n+1}$. Then $s_n = 1$ if $n$ is odd, and $s_n = 0$ if $n$ is even. $\frac{1}{N}\sum_{n \le N} s_n \to \frac{1}{2}$, so if $s_n \to \frac{1}{2}$ in some metric $d$ on $\mathbb{R}$, we must have $d(1,\frac{1}{2}) = d(0,\frac{1}{2}) = 0$, which doesn't happen.