# Converges Proof

1. Apr 28, 2010

### DEMJ

1. The problem statement, all variables and given/known data

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.

2. Relevant equations
It is true or false.

3. 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?

Last edited: Apr 28, 2010
2. Apr 28, 2010

### CompuChip

Looks okay.

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