I have trouble finding the limit, some help please? $\displaystyle \lim_{n \to \infty}\frac{(4n+2)n^n}{(n+1)^{n+1}}$
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You need to notice that: $\displaystyle \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} $
$\displaystyle \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}} $ $\displaystyle = \lim_{n \to \infty} \frac{4n+2}{n+1} \cdot \lim_{n \to \infty} \frac{n^{n}}{(n+1)^{n}} = \frac{4}{e}$
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