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In one of the lecture notes I've found that $C_n$ $$ C_n= \begin{cases} \frac{n!}{\sqrt 2 \Gamma((n/2+1)}\pi^{-1/42^{-n/2}(n!)^{-1/2}} & n\text{ even} \\[4mm] \frac{2(n!)}{(\sqrt2n+1/(\sqrt2 n))\Gamma(n+1/2)}\pi^{-1/42^{-n/2}(n!)^{-1/2}}, & n\text{ odd} \end{cases} $$ is essentially $(n!)^{-1/2}$ for any $n\in N$.

I was trying to estimate $C_n$ and I was getting $C_n\approx\dfrac{(n!)^{1/2}}{2^{n/2}}$.

I cannot find my mistake. Please help me to understand how come $C_n\approx (n!)^{-1/2}$.

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  • $\begingroup$ You used a relatively complicated "array" environment where a fairly simple "cases" environment gives better results. And somehow you left the expressions "$n\text{ even}$" and "$n\text{ odd}$ inside the superscripts. Also I set "even" and "odd" as \text{ even} and \text{ odd}, with the initial blank space inside \text{}, so that the spacing is taken care of that way. $\endgroup$
    – Michael Hardy
    May 24, 2012 at 17:37
  • $\begingroup$ @Michael Hardy: Thank you. $\endgroup$
    – David
    May 24, 2012 at 19:22

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