Does the Divergence of ∑b_n Imply ∑a_n Also Diverges?

[Bigworldjust]

Homework Statement

Suppose that ∑a_n and ∑b_n are series with positive terms and ∑b_n is divergent. Prove that if:

lim a_n/b_n = infinity
n--->infinity

then ∑a_n is also divergent.

Homework Equations

The Attempt at a Solution

Well in attempting to write a viable solution, I have deducted that since both series have positive terms, both sequences are increasing. If ∑b_n is is divergent and the limit as n approaches infinity of a_n/b_n is infinity than ∑a_n also must be divergent. Is there anymore to this however? I think I am missing something important in the explanation but I am not too sure of what it is. Thank you!

 

[lanedance]

I would start with your definition of divergnence, what is it?

Qualitatively, hopefully you can see what is going on the series bn diverges, but for some n>N, every term is an is much larger that the bn term hence the sum over an diverges

an example is:
$$b_n = \frac{1}{n}$$
$$a_n = \frac{1}{\sqrt{n}}$$