Consider the sum
$S(n) = \frac{1^{n-1}}{2^n} + \frac{2^{n-1}}{3^n} + \frac{3^{n-1}}{4^n} + \cdots \infty = \sum_{k=1}^\infty \frac{k^{n-1}}{(k+1)^n}$
How do I find the value of $\lim_{n\to\infty}S(n)$?
I am guessing it would be zero. But then again that's a guess! ;)