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 :)