Let be a positive and strictly decreasing sequence such that . Prove that the series

diverges.

**Solution**

**Lemma: **Let . It holds that

*Proof: *We are using induction on . For it is trivial. Suppose that it holds for . Then

Thus it holds for and the lemma is proved. Since and I can find such that for all . But then

where in the first inequality the lemma was used.

The exercise can also be found at mathematica.gr . It can also be found in Problems in Mathematical Analysis v1 W.J.Kaczor M.T.Nowak as exercise 3.2.43 page 80 .