# Alternating Series Test

• February 6th 2013, 04:43 AM
renolovexoxo
Alternating Series Test
* I hope its alright that I reposted this. I realized that Calculus might not be the best category for this. *

Show by example that the hypothesis b1>=b2>=...bn>=0 cannot be replaced by bk>=0 and limit k-->infinity =0

hint: use |ab|< 1/2(a2+b2)

I've found an example: bk=(1/k2 + 1/k) which satisfies the limit going to zer and all terms being positive, which diverges. I'm getting stuck with a rigorous proof of this, I thought about using the Dirichlet Test to prove this, but I'm getting hung up I think. Any help?
• February 6th 2013, 12:13 PM
johng
Re: Alternating Series Test
I'm afraid your counterexample is actually convergent. That is $\sum^\infty_{k=1}(-1)^{k+1}(1/k^2+1/k)$ is the sum of two convergent alternating series. I really don't see what example the hint is implying. However, here's an example and a few more comments.

Attachment 26884