limit

**dori1123** · January 31st 2010, 01:48 PM

I have trouble finding the limit, some help please?
$\lim_{n \to \infty}\frac{(4n+2)n^n}{(n+1)^{n+1}}$

**Plato** · January 31st 2010, 01:54 PM

You need to notice that:
$\frac{{n^n }} {{\left( {n + 1} \right)^n }} = \left( {\frac{{n + 1}} {n}} \right)^{ - n} = \left( {1 + \frac{1} {n}} \right)^{ - n} \to \frac{1} {e} $

**Random Variable** · January 31st 2010, 02:50 PM

$\lim_{n \to \infty} \frac{(4n+2)n^n}{(n+1)^{n+1}} = \lim_{n \to \infty} \frac{4n+2}{n+1} \frac{n^{n}}{(n+1)^{n}}$

$= \lim_{n \to \infty} \frac{4n+2}{n+1} \cdot \lim_{n \to \infty} \frac{n^{n}}{(n+1)^{n}} = \frac{4}{e}$

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