The series (1x2)/(3^2x4^2) + (3x4)/(5^2x6^2) + (5x6)/(7^2+8^2) +... is convergent?
a) True
b) False
Is the answer true?
Thanks!
Bruce
So the series is $\displaystyle \sum_{n=1}^\infty \frac{n(n+1)}{(n+2)^2(n+3)^2}$.
If you know the limit comparison test, you should be able to apply it directly.
With a little effort, you can use the comparison test:
$\displaystyle (n+2)^2 = n^2+4x+4 > n^2+4x = n(n+4)$
$\displaystyle (n+3)^2 = n^2+6x+9 > n^2+6x+5 = (n+1)(n+5)$
so
$\displaystyle \frac{n(n+1)}{(n+2)^2(n+3)^2} < \frac{n(n+1)}{n(n+4)(n+1)(n+5)} = \frac{1}{(n+4)(n+5)}$
and $\displaystyle \sum_{n=1}^\infty \frac{1}{(n+4)(n+5)}$ is a convergent series.
- Hollywood