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 2π.

Solution