# Integral test, basic comparsion test, limit comparsion test

1) http://www.geocities.com/asdfasdf23135/calculus02.JPG

This question comes from a section (infinite series) related to the integral test, basic comparsion test, and limit comparsion test, so I believe that I have to use one of them. However, I seriously have no idea how to prove this...can someone give me some hints or guidelines on how to solve this problem?

Thanks a lot!

HallsofIvy
Homework Helper
Compare ak/k with 1/k2!

But I don't know what a_k is, how can I know which of (a_k)/k and 1/k^2 is bigger?

And also, how can we make use of the fact that Sigma (a_k)^2 converges?

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I vaguely recall that the Cauchy-Schwarz inequality was useful in a problem similar to this. Or possibly the same problem. I forget.

I haven't learnt the Cauchy-Schwarz inequality in this course, is there any way to prove without it?

I don't know of any off the top of my head. Sorry. But you might look for versions that you could prove relatively easily from what you know. Said inequality comes in many different flavors.

use the limit comparison test...

if lim(n->infinity) of (a_k/k)/(a^2_k) = L > 0 (is greater than zero because a_k has all nonnegative terms) and sum of (a^2_k) converges (which it says it does), then sum of (a_k/k) must converge by the limit comparison test

lim(n->infinity) of (a_k/k)/(a^2_k)= lim of 1/(k*a_k) which is greater than zero
...if you didn't get the limit part

But lim(n->infinity) of (a_k/k)/(a^2_k) might not even exist, so how can you use the limit comparsion test?

Dick