Question

show that ( 1-(1/n) )^n < 1/e for all n

Answer 1

http://www.wolframalpha.com/input/?i=lim+n-%3Einf+%28+1-%281%2Fn%29+%29^n+

Answer 2

Since the series converge to 1/e, proving the following would be sufficient to prove that the sated relation holds true
\[ \left ( 1 - \frac{1}{n+1} \right )^{n+1} > \left ( 1 - \frac{1}{n} \right )^{n} \]
\[ n^{2n+1} > (n^2 - 1)^n(n+1)\]