Prove that the series $\sum\limits_{k=1}^{\infty}[\ln(ak+b)- \ln(ak)]$ diverges

Question

Let a and b be positive numbers.

Prove that the series $\sum_{k=1}^{\infty}(ln(ak+b)- ln(ak))$ diverges.

At first I thought expanding it would mean a few terms get cancelled out but it only works out for a few values of a and b. But then I realized that wouldn't work out since the sum is to infinity.

Any hints on how to approach this?

Answer 1

HINT:

Note that $\log(1+x)\ge \frac{x}{1+x}$. Then, we have

$$\log(ak+b)-\log(ak)=\log\left(1+\frac{b}{ak}\right)\ge \frac{b}{ak+b}$$