Puzzles/Analytical Puzzles/Surprising Limit/Hint

Puzzles|Analytical puzzles | Surprising Limit|Hint

Show that

$$\lim_{n\rightarrow\infty}n\sin(2\pi n!e)=2\pi.$$

(calculable in head!)

Hint

 * Use the series expansion of e:

$$e=1+\frac{1}{2}+\frac{1}{3!}+\frac{1}{4!}+...$$


 * $$sin$$ is a periodic function with periodicity $$2\pi$$.

Solution