# Thread: find the sume of a series

1. ## find the sume of a series

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

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

I note that by regrouping the terms cancel out.

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

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?

2. $= \frac{1}{2}+\left(-\frac{1}{3} + \frac{1}{3}\right)+...+\left(-\frac{1}{n-1} + \frac{1}{n-1}\right) - \frac{1}{n}=\frac{1}{2}+\frac{1}{n}=\frac{1}{2}$ when $n \to \infty$