Register to reply 
Convergent series with nonnegative terms, a counterexample with negative terms 
Share this thread: 
#1
Aug2311, 12:55 PM

P: 2

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

Mentor
P: 21,311




Register to reply 
Related Discussions  
QFT Counter Terms example calculation?  Quantum Physics  16  
Is there a reason the Ampere is defined in terms of negative charge per unit time?  Classical Physics  7  
Are all quadratic terms in gauge fields necessarily mass terms?  Quantum Physics  0  
Charge density in terms of (r,θ) but need it in terms of the vector r'  Advanced Physics Homework  8  
Passive sign convention (negative watts, and negative current confusion)  Advanced Physics Homework  7 