Sum of a Series: Finding the Convergence and Limit of a Series

[zeion]

Homework Statement

Find the sum of the series.

\sum_{k=0}^\infty \frac{1}{(k+1)(k+3)}

Homework Equations

The Attempt at a Solution

<br /> = \frac{1}{1\cdot3} + \frac{1}{2\cdot4} + \frac{1}{3\cdot5} + ... + \frac{1}{(n+1)\cdot(n+3)}<br />

<br /> = \frac{1}{2} [(1-\frac{1}{3}) + (\frac{1}{2} - \frac{1}{4}) + (\frac{1}{3} - \frac{1}{5}) + ... + (\frac{1}{(n+1)} - \frac{1}{(n+3)})<br />

<br /> = \frac{1}{2}[ 1 + (\frac{1}{2} + \frac{1}{3} + \frac{1}{n+1}) - (\frac{1}{3} + \frac{1}{4} + \frac{1}{5} + ... + \frac{1}{n+3}) ]<br />

So here
<br /> (\frac{1}{2} + \frac{1}{3} + \frac{1}{n+1}) \to 1<br />

<br /> (\frac{1}{3} + \frac{1}{4} + \frac{1}{5} + ... + \frac{1}{n+3}) \to 0<br />

Then the whole thing sums to 1?

 

[JSuarez]

Ok, things went very bad from the second to the third line in your argument; why don't you try to cancel the terms with opposite signs, instead of grouping them?