Does the sum of ln(k/(k+1)) converge or diverge as n approaches infinity?

[cp255]

So I was trying to see if \Sigmaln(\frac{n}{n+1}) diverges or converges. To see this I started writing out [ln(1) - ln(2)] + [ln(2) - ln(3)] + [ln(4) - ln(5)] ...

I noticed that after ln(1) everything must cancel out so I reasoned that the series must converge on ln(1) which equals ZERO. However, Wolfram Alpha says the series is divergent. I tried looking it up how to do this problem and I read that the last term also does not cancel out which is why the series diverges. However, shouldn't there always be a term bigger than the last so everything must cancel.

 

[Infrared]

Write an expression for \sum_{k=1}^{n} ln(\frac{k}{k+1}). What is the limit of this expression as n tends to infinity?