Originally Posted by

**icelated** I am trying to confirm the integral test can be applied. I just cant see how the solution manual gets there solution.

I think there using the quotient rule but i cant seem to work it out.

Can someone help me work out the problem?

$\displaystyle \sum_{n=1}^\infty \frac 1 {\sqrt {n} ( \sqrt {n} + 1 )} $

then,

Let $\displaystyle f(x) = \frac 1 {\sqrt {x} ( \sqrt {x} + 1 )} $

The solution manual then shows!

$\displaystyle f'(x) = \frac {1+2\sqrt x} { 2x^{3/2} ( \sqrt x + 1 )^2} < 0$

With the denominator squared it does show they use the quotient rule. However, can you please

show me the steps on how they derive the answer?

Because i cant seem to get it. Its kinda a nasty problem.

Thank you so very much!