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Convergence of a series

  • Thread starter sitia
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  • #1
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Homework Statement


Consider sequences {an} and {bn}, where sequence {bn} converges to 0.
Is it true that sequence {anbn} converges to 0?



The Attempt at a Solution



Proof.
First I assumed (an) is bounded, and so there exists M > 0 such that |an| < M for all n 2
{1, 2, 3, . . .}. Moreover, since lim bn = 0, there exists N such that, for all n≥ N,
|bn − 0| <ε /M
or, equivalently, |bn| < ε/M.
Therefore, whenever n ≥ N, we have that
|anbn − 0| = |anbn| = |an||bn| ≤M|bn| < M(ε/M)=ε

Since the choice of  ε> 0 was arbitrary, this implies that lim(anbn) = 0.

However, I know that {anbn} can also not converge if {an} is not bounded. Can someone help with how to go about that part of hte proof?
Thank you!
 

Answers and Replies

  • #2
It obviously doesn't work when an is not bounded. I am understanding correctly that an does not converge to 0, right?
if so, just consider bn=1/n and an=n^2. an*bn=n does not convergen. Bam!
 
  • #3
25
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There isn't given any information on if {an} converges to 0 or not. Thanks
 

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