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Convergent series with nonnegative terms, a counterexample with negative terms 
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#1
Aug2311, 12:55 PM

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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 nonnegative. 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 counterexample 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 counterexample. 


#2
Aug2311, 01:29 PM

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P: 21,216




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