Testing the Convergence of Series: A Counterexample

[DEMJ]

Homework Statement

If \sum_{k=1}^{\infty} a_k converges and a_k/b_k \to 0 as k\to \infty, then \sum_{k=1}^{\infty} b_k converges.

Homework Equations

It is true or false.

The Attempt at a Solution

I think it is false and here is my counterexample. Let a_k = 0,b_k=\frac{1}{k}. This satisfies our initial conditions of \sum_{k=1}^{\infty} a_k converges and a_k/b_k \to 0 as k\to \infty but \sum_{k=1}^{\infty} b_k diverges.
Is this correct?

 

[CompuChip]

Looks okay.

Your counterexample also looks correct, if you want to make it slightly less trivial you could use
a_k = \frac{1}{k^2}
instead :)