Infinitie Series: Converge/Diverge question

[trap]

this question doesn't look hard when i read it, but then when i tried doing it, i can't

Suppose $$\sum$$Ak and $$\sum$$Bk are both convergent series with positive terms. Does it follow that $$\sum$$AkBk converges? Show your answer is correct with an appropriate proof/ counter example.

Thanks for help.

 

[matt grime]

Yes, it converges. Here's the hint:

since sum A_n converges, A_n tends to zero, hence for all k greater than some N, 0<=A_n< 1

now apply some comparison test to reach the conlcusion.


to make it more interesting how about:

Secondly, use the Cauchy schwartz inequality to prove the result (hint, you'll need to show sum (A_n)^2 exists.

Thirdly, show that if we drop the positivity conditions on the A_n and B_m then the result is false (hint alternating series test - if C_m is a sequence of positive terms tending to zero then sum (-1)^mC_m converges)

 

[mathman]

Since the tails of both the A series and the B series must ->0, the tail of the product series goes to 0 faster. To put it rigorously, there exists a K such that for all k>K, A_kB_k<A_k, therefore the series of products must converge.

 

[matt grime]

You're not using the positivity condition, which is required. Though that is a small point.

 

[mathman]

(matt grime) I presume you remark was addressed to me. I didn't use the words. However, I know the positive condition was necessary, and I didn't bother using | | for the terms involved.