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## Homework Statement

Find the sum of the series.

[tex] \sum_{k=0}^\infty \frac{1}{(k+1)(k+3)} [/tex]

## Homework Equations

## The Attempt at a Solution

[tex]

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

[/tex]

[tex]

= \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)})

[/tex]

[tex]

= \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}) ]

[/tex]

So here

[tex]

(\frac{1}{2} + \frac{1}{3} + \frac{1}{n+1}) \to 1

[/tex]

[tex]

(\frac{1}{3} + \frac{1}{4} + \frac{1}{5} + ... + \frac{1}{n+3}) \to 0

[/tex]

Then the whole thing sums to 1?