This is a homework question and I am not really sure where to go with it. I have a lot of trouble with sequences and series, can I get a tip or push in the right direction?
[Math] Prove that $(1 – \frac{1}{n})^{-n}$ converges to $e$
sequences-and-series
Best Answer
You have: $$ x_n:=\left(1-\frac1n\right)^{-n} = \left(\frac{n-1}n\right)^{-n} = \left(\frac{n}{n-1}\right)^{n} $$ $$ = \left(1+\frac{1}{n-1}\right)^{n} = \left(1+\frac{1}{n-1}\right)^{n-1}\cdot \left(1+\frac{1}{n-1}\right) = a_n\cdot b_n. $$ Since $a_n\to \mathrm e$ and $b_n\to 1$ you obtain what you need.