Convergence of $1 - \sqrt{ \frac{2}{n-1} } \cdot \frac{ \Gamma(n/2) }{ \Gamma( \frac{n-1}{2} ) } $ as $n \to \infty$

Question

I stumbled upon this problem while reading about the bias of the sample standard deviation.

How to show that: $$\bigg(1 - \sqrt{ \frac{2}{n-1} } \cdot \frac{ \Gamma(n/2) }{ \Gamma( \frac{n-1}{2} ) } \bigg) \sim \frac{1}{4 n}$$ as $n \to \infty$ and n is positive integer.

I thought I should use Stirling's formula, or Gautschi's inequality. The following equality might also lead in the right direction: $$\Gamma(1/2+n)=\frac{(2n)!}{4^nn!}\sqrt{\pi}$$

Can you help?

Answer 1

By Stirling's formula we have

$$1 - \sqrt{ \frac{2}{n-1} } \cdot \frac{ \Gamma(n/2) }{ \Gamma( \frac{n-1}{2} ) } \sim 1 - \sqrt{ \frac{2}{n-1} } \cdot \frac{ \sqrt{2\pi \left(\frac n 2 -1\right)}\left(\frac {\frac n 2 -1} e\right)^{\frac n 2 -1} }{\sqrt{2\pi \left(\frac n 2 -\frac 32\right)}\left(\frac {\frac n 2 -\frac 32} e\right)^{\frac n 2 -\frac32} } $$

$$=1 - \sqrt{ \frac{2}{n-1} } \cdot \frac1{\sqrt{2e}}\frac{n-3}{\sqrt{n-2} }\left(\frac{n-2}{n-3}\right)^\frac n 2 = \ldots$$

and since

$$\sqrt{ \frac{2}{n-1} } \cdot \frac{n-3}{\sqrt{n-2} }=\sqrt 2 \sqrt{1-\frac{3n-6}{n^2-3n+3}}=\sqrt 2 \left(1-\frac 3{2n}+O\left(\frac1{n^2}\right)\right)$$

$$\left(\frac{n-2}{n-3}\right)^\frac n 2= \left(1+\frac{1}{n-3}\right)^\frac n 2= \sqrt e\left(1+\frac 5{4n}+O\left(\frac1{n^2}\right)\right)$$

we have

$$\ldots = 1-\left(1-\frac 3{2n}+O\left(\frac1{n^2}\right)\right)\left(1+\frac 5{4n}+O\left(\frac1{n^2}\right)\right)=1-\left(1-\frac 3{2n}+\frac 5{4n}+O\left(\frac1{n^2}\right)\right)=\frac1{4n}+O\left(\frac1{n^2}\right)$$

Answer 2

Using twice Stirling formula, we have $$\log \left(\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma \left(\frac{n-1}{2}\right)}\right)=\frac{1}{2} \log \left(\frac{n}{2}\right)-\frac{3}{4 n}-\frac{1}{2 n^2}-\frac{3}{8 n^3}+O\left(\frac{1}{n^4}\right)\tag 1$$ Using the binomial expansion and taking the logarithm and ussing Taylor series $$\log \left(\sqrt{\frac{2}{n-1}}\right)=-\frac{1}{2} \log \left(\frac{n}{2}\right)+\frac{1}{2 n}+\frac{1}{4 n^2}+\frac{1}{6 n^3}+O\left(\frac{1}{n^4}\right)\tag 2$$ Adding $(1)$ to $(2)$ gives $$\log \left(\sqrt{\frac{2}{n-1}}\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma \left(\frac{n-1}{2}\right)}\right)=-\frac{1}{4 n}-\frac{1}{4 n^2}+O\left(\frac{1}{n^3}\right)\tag 3$$ $$\sqrt{\frac{2}{n-1}}\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma \left(\frac{n-1}{2}\right)}=1-\frac{1}{4 n}-\frac{7}{32 n^2}+O\left(\frac{1}{n^3}\right)\tag 4$$ $$1-\sqrt{\frac{2}{n-1}}\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma \left(\frac{n-1}{2}\right)}=\frac{1}{4 n}+\frac{7}{32 n^2}+O\left(\frac{1}{n^3}\right)\tag 5$$

Try for $n=10$; the exact value is $$1-\frac{128 }{105}\sqrt{\frac{2}{\pi }}=0.0273407\cdots$$ while the truncated series gives $$\frac{87}{3200}=0.0271875\cdots$$