<s>[INST] Asymptotic expression to $e^{-k^2/2n}$ I want to show that $$\Bigl(1-{k\over n}\Bigr)^{-(1+n-k)/2}\Bigl(1+{k\over n}\Bigr)^{-(1+n+k)/2}$$ is asymptotic to $e^{-k^2/2n}$.
I, in my earlier post, was satisfied with the given answer. But only now I noticed that the answer provided has a flaw since though it claims to produce $e^{-k^2/2n}$, it actually produces $e^{-k^2/n}$.
Any thoughts?
 [/INST] You have 
\begin{align}
\Bigl(1-{\tfrac kn}\Bigr)^{-\frac{1+n-k}2}\Bigl(1+{\tfrac kn}\Bigr)^{-\frac{1+n+k}2}
&=
\exp\left[-\tfrac{1+n-k}2\,\log\left(1-\tfrac kn\right)-\tfrac{1+n+k}2\,\log\left(1+\tfrac kn\right)\right]\\ \ \\
&=\exp\left[-\tfrac{1+n-k}2\,\left(-\tfrac kn-\tfrac{k^2}{2n^2}+o(\tfrac{k^3}{n^3})\right)-\tfrac{1+n+k}2\, \left(\tfrac kn-\tfrac{k^2}{2n^2}+o(\tfrac{k^3}{n^3})\right)\right]\\ \ \\
&=\exp\left[-\tfrac{1+n-k}2\,\left(-\tfrac kn-\tfrac{k^2}{2n^2}+o(\tfrac{k^3}{n^3})\right)-\tfrac{1+n+k}2\, \left(\tfrac kn-\tfrac{k^2}{2n^2}+o(\tfrac{k^3}{n^3})\right)\right]\\ \ \\
&=\exp\left[-\tfrac{k^2}n+(1+n)\tfrac{k^2}{2n^2}+o(\tfrac1{n^2}) \right]\\ \ \\
&=\exp\left[-\tfrac{k^2}{2n}+\tfrac{k^2}{2n^2}+o(\tfrac1{n^2}) \right]\\ \ \\
&=\exp\left[-\tfrac{k^2}{2n}+o(\tfrac1{n^2}) \right]\\ \ \\
\end{align}
</s>