- #1
zeion
- 466
- 1
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?