Here's a challenge problem I just made up:
Let be a sequence of positive real numbers such that . Show that
.
Let .
We want to show that diverges as .
( I chose this notation to suggest the similarity with , if )
We assume that otherwise the series would diverge trivially.
Let , then
We observe that diverges since and now by the limit comparison test - both differences are always positive-, we get that diverges too.
Notation:
Your solution is same as what came to my mind when I saw the question.
For those who might want to see a reference for the relation between the sum and the product used by Bruno J, please see [Theorem 7.4.6, 1].
I quote the result for convenience.
Lemma. Let be a sequence of nonnegative numbers, then
and converge or diverge together.
References
[1] L.S. Hahn and B. Epstein, Classical Complex Analysis, Jones and Bartlett Publishers, Inc., London, 1996.