plugging index values in.Find the some of $\displaystyle S = \sum_{i = 1}^{\infty} (\frac{1}{i + 1} - \frac{1}{i + 2})$

$\displaystyle (\frac{1}{2} - \frac{1}{3}) + (\frac{1}{3} -\frac{1}{4} ) + (\frac{1}{4} -\frac{1}{5} )$

I note that by regrouping the terms cancel out.

$\displaystyle \frac{1}{2} + (- \frac{1}{3} + \frac{1}{3} ) + (-\frac{1}{4} + \frac{1}{4} ) -\frac{1}{5} . . .$

and looks like the sum is 1/2 but I still have the tail and I have not show how it goes to 0 as n goes to infinity. how would i phrase the tail to get a limit resulting in zero?