True of false: The sum of this infinite series. I'm fairly sure it is false, but I'm not quite sure about which test I should use to prove it.
$$\sum_{n=2}^\infty \ln\left(\frac{n-1}{n}\right) = -1 $$
I think using the integral test should work, but it may get kind of messy, so I'm looking for som advice. All I have to do is prove that it diverges, right? 
Thanks in advance.

It is actually very easy. See that
$$f(N)=\sum_{n=2}^N\ln \left(\frac{n-1}{n}\right)=
\sum_{n=2}^N\ln \left(1-\frac{1}{n}\right)$$
Is decreasing.
Now $f(3)\simeq-1.09861>f(+\infty)$. So $f(+\infty)\neq -1$.
