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

[Crossfader]

Homework Statement

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

Prove that $\sum_{n=1}^\infty$$m_n$ converges.

Show with a counter-example that the claim above doesn't necessarily hold if the assumption $a_n$$\geq$0 for all n$\geq$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 $m_n$=max{$a_n, a_{n+1}$}

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

 

[Mark44]

Homework Statement

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

Prove that $\sum_{n=1}^\infty$$m_n$ converges.

Show with a counter-example that the claim above doesn't necessarily hold if the assumption $a_n$$\geq$0 for all n$\geq$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 $m_n$=max{$a_n, a_{n+1}$}

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

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?