# Math Help - about sequences 1

Hello,

Show that $lim_{n\rightarrow +\infty}n[1-\frac{(n+1)^{n}}{en^{n}}]=\frac{1}{2}.$

2. Originally Posted by Biscaim
Hello,

Show that $lim_{n\rightarrow +\infty}n[1-\frac{(n+1)^{n}}{en^{n}}]=\frac{1}{2}.$

It looks to me like you'll need to develop this expression as a series. See - Wolfram|Alpha

3. Originally Posted by Biscaim
Hello,

Show that $lim_{n\rightarrow +\infty}n[1-\frac{(n+1)^{n}}{en^{n}}]=\frac{1}{2}.$

$n[1-\frac{(n+1)^n}{en^n}] = \frac{1 - \frac{1}{e}\Big(1+\frac{1}{n}\Big)^n}{\frac{1}{n}}$. By definition of e, the numerator approaches 1 - 1/e * e = 0, so maybe applying L'Hopital's rule will be successful (I haven't tried it).