How do you prove that the infinite sum   [math]\displaystyle\sum_{k=1}^{\infty}(n+1)^{-k}[/math]    converges to [math]\dfrac{1}{n}?[/math]
How can you show that the infinite sum [math]\sum_{n=1}^\infty \frac{n}{2^{n+1}}[/math] is 1?