Manteo2h

2022-06-27

Is this : convergent sum?

humusen6p

Expert

One has, as $n\to \mathrm{\infty }$,
$\frac{\mathrm{tan}\frac{1}{n!}}{\mathrm{arctan}\left(n!\right)}=\frac{\mathrm{tan}\frac{1}{n!}}{\frac{\pi }{2}-\mathrm{arctan}\frac{1}{n!}}\sim \frac{2}{\pi }\cdot \frac{1}{n!}$
then by the comparison test the series $\sum _{n\ge 1}\frac{\mathrm{tan}\frac{1}{n!}}{\mathrm{arctan}n!}$ is convergent.

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