L'Hopital's rule and series convergence

sequences-and-series analysis convergence-divergence calculus

Take $f(x) = \log \log x,$ then $g(x) = x \log x.$ The sum of $1/g$ diverges, so the sum of $f/g$ also diverges. But $f' / g'$ is slightly smaller than $$ \frac{1}{x (\log x)^2} $$ and this sum converges.

For this, you need to notice that an antiderivative of $\frac{1}{x \log x}$ is $\log \log x,$ while an antiderivative of $\frac{1}{x (\log x)^2}$ is $\frac{-1}{\log x}.$

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