Does $\sum \limits_{n=2}^\infty n$ have a greater Cesàro mean than $\sum \limits_{n=1}^\infty n$?
If not, then is there any other sense of "mean" in which the former's mean is greater than the latter's?
(Sorry if this question makes no sense. I have no idea what I'm doing.)
Both have no limit, so one could say, they are both infinite. And whether $\infty-1 \ne \infty$ makes some sense at all is up to some, I'd say, very specific context - if such context exists at all...
Note, that the Cesaro-summing gives "means" in the sense, that the partial means of the partial sums converge to some fixed number. So it is most used in the case of alternating signs, for instance $s_1 \underset{\mathcal {Ces}}{=}\sum_{n=1}^\infty (-1)^n n $, and $s_2 \underset{\mathcal {Ces}}{=} \sum_{n=2}^\infty (-1)^n n $ . And then indeed we arrive at $s_1 \ne s_2$