Analize convergence of an infinite series

This is the series:

$\displaystyle \sum_{n=1}^{\inf}{ (\tfrac{1}{2n} - \tfrac{1}{2(n+1)})}$

So this is what I did:

$\displaystyle \sum_{n=1}^{\inf}{ (\tfrac{1}{2n} - \tfrac{1}{2(n+1)})} = \sum_{n=1}^{\inf}{ \tfrac{1}{2n}} - \sum_{n=1}^{\inf} {\tfrac{1}{2(n+1)}}=

\tfrac{1}{2} (\sum_{n=1}^{\inf}{ \tfrac{1}{n}} - \sum_{n=1}^{\inf} {\tfrac{1}{(n+1)})}$

So the first term diverges because it is harmonic and thus the hole series diverges.

I have this marked as wrong, but I don't understand why. Can somebody explain it please?