# Convergence of a series..Watch

#1
with for all. If converges, then show that converges.

I was thinking of using ratio test conditions here, btu the that sort of falls apart if we have the b_n series converging but the ratio tending to 1. Any ideas how I should go about this..
0
8 years ago
#2
There is the other fact that a series sum a_n can converge while the limit a_n+1 / a_n dosen't exist.

If sum b_n converges then b_n tends to 0...
0
8 years ago
#3
.

Can you see where to go now?
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#4
(Original post by around)
.

Can you see where to go now?
Unfourtunately, not really. I know i need to get an upper bound on a_n to do with b_n or something about b so that someting to do with the b series bounds the a series and so it converges.

I'm not sure because as Deank22 said, b_n+1/b_n may not converge and so it may not be bounded...
0
8 years ago
#5
When faced with a sum it's always a good idea to write out a few terms:

We know b_n converges, so it's a good idea to somehow bound the terms of our sequence above by the terms of the b_n sequence. Using the fact that , we can rewrite the first two terms:

We can also apply inductively: .

You can finish this off now.
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