# Math Help - Limit

1. ## Limit

Hi!
How do I show that
$\lim_{n\to\infty}\int_0^1 \frac{1}{\sqrt{1-x^n}} \, dx =1$ ?

I've proved that
$\int_0^1 \frac{1}{\sqrt{1-x^n}} \, dx=\frac{\sqrt{\pi}}{n} \frac{\Gamma(\frac{1}{n})}{\Gamma(\frac{1}{n}+\fra c{1}{2})}$

Thanks!

2. Originally Posted by DavidEriksson
Hi!
How do I show that
$\lim_{n\to\infty}\int_0^1 \frac{1}{\sqrt{1-x^n}} \, dx =1$ ?

I've proved that
$\int_0^1 \frac{1}{\sqrt{1-x^n}} \, dx=\frac{\sqrt{\pi}}{n} \frac{\Gamma(\frac{1}{n})}{\Gamma(\frac{1}{n}+\fra c{1}{2})}$

Thanks!
It is fact that $2^{2x-1} \Gamma(x + \frac{1}{2} ) = \frac{ \sqrt{\pi}\Gamma(2x) }{\Gamma(x)}$ Try to use this to finish it .