Thread: a critical point(?) to a sum of series being convergent or divergent ;;

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 ???

Help! ><

Discuss the cases and .

AHA! wow!!! I didn't even realize that the question is related to the zeta function! z(s)
When a <= 1, it diverges and when a>1, it converges... now i get it
Thanks! =)