# Error estimate for sequence defining e

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, $$e-\left(1+\frac1n\right)^n=\frac{e}{2n}+O\left(\frac1{n^2}\right)$$