Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
add comment

1 Answer

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

share|improve this answer
Awesome. Thnaks! –  Amir Feb 5 '13 at 0:16
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.