Let $n\geq1$. Find a formula for the sum:
$$S_n=\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\cdots+\frac{1}{n\cdot(n+1)}$$
The formula for the sum should be based on n components of the last term:
$$S_n= n\cdot\frac{1}{n(n+1)}$$
So that should give:
$$\sum_{n=1}^\infty\frac{1}{(n+1)}$$
Proof by induction, consider the base case $n=1$:
$$\sum_{n=1}^1\frac{1}{(1+1)}=\frac{1}{2}$$
But how do I prove the case for $n+1$? Thanks