The question:

$\displaystyle \sum _{n=2}^{\infty }{\frac {1}{n \left( \ln \left( n \right)

\right) ^{2}}}$

I need help in finding if $\displaystyle f \left( x \right) ={\frac {1}{x \left( \ln \left( x \right)

\right) ^{2}}}

$ is decreasing.

I found that $\displaystyle {\frac {d}{dx}}f \left( x \right) =-{\frac { \left( \ln \left( x

\right) \right) ^{2}+2\,\ln \left( x \right) }{{x}^{2} \left( \ln

\left( x \right) \right) ^{4}}}$. Then I made $\displaystyle -{\frac { \left( \ln \left( x \right) \right) ^{2}+2\,\ln \left( x

\right) }{{x}^{2} \left( \ln \left( x \right) \right) ^{4}}}<0

$

which turns out to be $\displaystyle {e}^{-2}<x$

but when I graph it, f(x) is still increasing at $\displaystyle {e}^{-2}$. What am I doing wrong?