Convergent series with non-negative terms, a counter-example with negative terms

Crossfader

1. The problem statement, all variables and given/known data

The terms of convergent series [itex]\sum_{n=1}^\infty[/itex][itex]a_n[/itex] are non-negative. Let [itex]m_n[/itex] = max{[itex]a_n, a_{n+1}[/itex]}, [itex]n = 1,2,...[/itex]

Prove that [itex]\sum_{n=1}^\infty[/itex][itex]m_n[/itex] converges.

Show with a counter-example that the claim above doesn't necessarily hold if the assumption [itex]a_n[/itex][itex]\geq[/itex]0 for all n[itex]\geq[/itex]1 is dropped.


2. The attempt at a solution

I think I've solved the first claim using a theorem which claims if series converges then its partial sum converges as well. This holds assuming that I understood right the meaning of [itex]m_n[/itex]=max{[itex]a_n, a_{n+1}[/itex]}

I'm stuck with another one, frankly saying I couldn't find any counter-example.

Mark44

How can a series fail to converge? One way is if the partial sums get larger and larger without bound (or more and more negative). Is there another way that a series can diverge?