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**1. Homework Statement**

show with mathematical induction that prod(k=1->n) (1+1/n)^n=(n+1)^n/n!

n element Natural numbers

**3. The Attempt at a Solution**

works for n=1,

[tex]\prod_{k=1}^{n+1}\left(1+\frac{1}{k+1}\right)^{k+1}[/tex] should become [tex]\frac{(n+2)^{n+1}}{(n+1)!}[/tex]

[tex]= (1+(n+1)^{-1})*((1+(n+1)^{-1})^n)*\frac{(n+1)^n}{n!}[/tex]

[tex]=\frac{(n+1)^n}{n!})*(1+(n+1)^{-1})(1+(n+1)^{-n})[/tex]

[tex]=\frac{(n+1)^n}{n!}*(1+(n+1)^n+(n+1)^{-1}+(n+1)^{-n+1}[/tex]

[tex]=\frac{(n+1)^n}{n!}+\frac{1}{n!}+\frac{(n+1)^n}{(n+1)!}+\frac{(n+1)^n}{(n+1)^{n+1})n!}[/tex]

[tex]=\frac{(n+1)^{n+1}+(n+1)+(n+1)^n+1)}{(n+1)!}[/tex]

only i can't find any way to get to (n+2)^(n+1)/(n+1)! from there :(

can anyone help?

thanks in advance :)

ps. Happy New Year to those it applies t

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