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


by Crossfader
Tags: convergent, counterexample, negative, nonnegative, series, terms
Crossfader
Crossfader is offline
#1
Aug23-11, 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 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.
Phys.Org News Partner Science news on Phys.org
Cougars' diverse diet helped them survive the Pleistocene mass extinction
Cyber risks can cause disruption on scale of 2008 crisis, study says
Mantis shrimp stronger than airplanes
Mark44
Mark44 is offline
#2
Aug23-11, 01:29 PM
Mentor
P: 21,059
Quote Quote by Crossfader View Post
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 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.
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?


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