<s>[INST] Which one is larger? I want to prove that for positive integer $n$,
$$\left(1-\frac{1}{n}\right)^n \left(1+\frac{1}{n}\right)^n \geq 1-\frac{1}{n}$$
but I am stuck on how to proceed. Can someone help me?
 [/INST] Consider
$$a_n=\left(1-\frac{1}{n}\right)^n \left(1+\frac{1}{n}\right)^n=\left(1-\frac{1}{n^2}\right)^n\implies \log(a_n)=n \log\left(1-\frac{1}{n^2}\right)$$ So, by Taylor
$$\log(a_n)=n\left(-\frac{1}{n^2}-\frac{1}{2 n^4}-\frac{1}{3
   n^6}+O\left(\frac{1}{n^8}\right)\right)=-\frac{1}{n}-\frac{1}{2 n^3}-\frac{1}{3
   n^5}+O\left(\frac{1}{n^7}\right)$$
$$a_n=e^{\log(a_n)}=1-\frac{1}{n}+\frac{1}{2 n^2}-\frac{2}{3 n^3}+\frac{13}{24 n^4}++O\left(\frac{1}{n^5}\right)$$ and, as Ross Millikan commented, this is an alternating series and then ...
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