Puzzles/Analytical Puzzles/Surprising Limit/Hint

Puzzles|Analytical puzzles | Surprising Limit|Hint

Show that

$\underset{n\to \mathrm{\infty }}{\lim }n\sin (2\pi n!e)=2\pi .$

(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