Testing the Convergence of Series: A Counterexample

[DEMJ]

Homework Statement

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

Homework Equations

It is true or false.

The Attempt at a Solution

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

 

[CompuChip]

Looks okay.

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