" alt="\displaystyle\sum_{r=1}^{\infty}\log\dfrac{r+1}{r} =\displaystyle\sum_{r=1}^{\infty}(\;\log({r+1})-\log {r}\" />

(telecopic series)

Fernando Revilla

Edited: Sorry, I didn't read the first part. So, an alternative is:

and the harmonic series is divergent, so the given series is divergent.