# Thread: How do I prove this is decreasing?

1. ## How do I prove this is decreasing?

$
\frac{1}{\sqrt{k(k + 1)}}
$

This is part of an alternating series and I need to prove that it converges conditionally. The only condition I am stuck on is proving that it's decreasing.

2. Originally Posted by TYTY
$
\frac{1}{\sqrt{k(k + 1)}}
$

This is part of an alternating series and I need to prove that it converges conditionally. The only condition I am stuck on is proving that it's decreasing.
$f(x) = (x^2 + x)^{-.5} f'(x)= -.5 (x^2 + x)^{-1.5} (2x + 1) x > 0$
so clearly the first derivative is negative, hence decreasing.

3. Originally Posted by apcalculus
$f(x) = (x^2 + x)^{-.5} f'(x)= -.5 (x^2 + x) (2x + 1) x > 0$
so clearly the first derivative is negative, hence decreasing.
Chain rule ftw