1 + 2 + 3 + 4 + ⋯

Remarks

It is not summable by Cesàro summation, since the averages of $s_N$ also tend to $+\infty$. It is not Abel summable either: for $0 < x < 1$, $\sum_{n \ge 1} n x^n = x/(1-x)^2$, which tends to $+\infty$ as $x \to 1^-$. Zeta function regularization assigns it the value $-1/12$. For $\operatorname{Re} s > 1$, $\sum_n n^{-s} = \zeta(s)$, where $\zeta$ is the Riemann zeta function, and the series of the example is formally the case $s = -1$; the analytic continuation of $\zeta$ has $\zeta(-1) = -1/12$. Ramanujan summation also gives $-1/12$. The value $-1/12$ is not a sum in the usual sense: the partial sums are all positive and grow without bound.