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Is there an estimate for error of $e-\left(1+\frac1n\right)^n$ ?thanks

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The power series for $\log(1+x)$ yields $$ n\log\left(1+\frac1n\right)=1-\frac1{2n}+O\left(\frac1{n^2}\right) $$ applying $e^{x+y}=e^xe^y$ and the power series for $e^x$ gives $$ \left(1+\frac1n\right)^n=e\left(1-\frac1{2n}+O\left(\frac1{n^2}\right)\right) $$ Therefore, $$ e-\left(1+\frac1n\right)^n=\frac{e}{2n}+O\left(\frac1{n^2}\right) $$

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Awesome. Thnaks! –  Amir Feb 5 '13 at 0:16

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