Answer
Divergent
Work Step by Step
$\Sigma^{\infty}_{n=1} \ln (\frac{n}{n+1}) = \Sigma^{\infty}_{n=1}(\ln(n) - \ln(n+1)) = S$
$S_{1}= \ln(n) = \ln 1, \ln 2, \ln 3,..., \ln (\infty -1)$
$S_{2}= -\ln(n+1) = -\ln2, -\ln3,...,-\ln(\infty-1), -\ln\infty$
$S_{1} + S_{2} = \ln1 -\ln\infty = -\infty$
$\lim\limits_{n \to \infty}S=-\infty$
Hence, the series diverges.
