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

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

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.
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Mentor
 Quote by Crossfader 1. The problem statement, all variables and given/known data 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?

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