Thread: Convergent Sequences, Harmonic Series

Prove or give a counterexample: Let an be a sequence such that the sequence bn=an+1 - an converges to 0. Does an have to converge?

Now I know this is not true and I know that it has something to do with partial sums of the harmonic series, but it's been years since I've worked with this sort of material and I am completely lost on how to use the harmonic series to disprove the theorem.

The solution is in fact to consider the partial sums of the harmonic series as the terms of the sequence.

Let . Since the harmonic series diverges, a_n diverges.
Clearly, and hence converges to 0.