What is the formula for $1/(1\cdot 2)+1/(2\cdot 3)+1/(3\cdot 4)+\ldots +1/(n(n+1))$

Question

How can I find the formula for the following equation?

$$\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\ldots +\frac{1}{n(n+1)}$$

More importantly, how would you approach finding the formula? I have found that every time, the denominator number seems to go up by n+2, but that's about as far as I have been able to get:

$1/2 + 1/6 + 1/12 + 1/20 + 1/30...$ the denominator increases by $4,6,8,10,12...$ etc.

So how should I approach finding the formula? Thanks!

Answer 1

If you simplify your partial sums, you get $\frac12,\frac23,\frac34,\frac45,....$ Does this give you any ideas?

Answer 2

Hint: Use the fact that $\frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}$ and find $S_n=\sum_1^n\left(\frac{1}{k}-\frac{1}{k+1}\right)$.