- #1
Crossfader
- 2
- 0
Homework Statement
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.