Well, I got curious about this...
It is quite easy to show that 1/1 + 1/2 + 1/3 + 1/4 + ... diverges to the positive infinity
and also, the fact that 1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + ... converges to a value pi^2/2, a.k.a basel problem is widely known.
Then, will there be a 'critical point(?)' s.t.
lim(n -> infinity) sigma[1/(n^a)]
(i.e. 1/1^a + 1/2^a + 1/3^a +1/4^a + ...)
be convergent or divergent ???