1 + 1 + 1 + 1 + ⋯

Remarks

The Cesàro means $(s_1 + \cdots + s_N)/N = (N+1)/2$ also tend to $+\infty$. For $0 < x < 1$, $\sum_{n \ge 1} x^n = x/(1-x)$, which tends to $+\infty$ as $x \to 1^-$, so the series is not Abel summable. Zeta function regularization assigns it the value $-1/2$. Indeed $\sum_n n^{-s} = \zeta(s)$ for $\operatorname{Re} s > 1$, the series of the example is formally the case $s = 0$, and the analytic continuation of the Riemann zeta function has $\zeta(0) = -1/2$. No linear and stable method can give it a finite value $S$: stability gives $S = 1 + S$, which is impossible.