tripes3h

2022-07-12

Does $\sum _{n=2}^{\mathrm{\infty }}\left(n\mathrm{ln}n{\right)}^{-1}$ diverge?
This seems like elementary calculus, but I can't figure this out. Can anyone supply a hint?

amanhantmk

Expert

HINT: Use the integral test: what happens to
${\int }_{2}^{\mathrm{\infty }}\frac{dx}{x\mathrm{ln}x}\phantom{\rule{thickmathspace}{0ex}}?$

Ciara Mcdaniel

Expert

Hint.
$\sum _{n={2}^{N}}^{{2}^{N+1}-1}\frac{1}{n\mathrm{ln}n}\ge \frac{{2}^{N}}{\left(N+1\right){2}^{N+1}}=\frac{1}{2\left(N+1\right)}$

Do you have a similar question?

Recalculate according to your conditions!