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

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- Feb 28th 2013, 04:25 AMweijing85Converence Sequence
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 - Feb 28th 2013, 06:38 AMhollywoodRe: Converence Sequence
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