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Math Help - Limit

  1. #1
    Junior Member
    Joined
    Aug 2009
    From
    Gothenburg, Sweden
    Posts
    37

    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!
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  2. #2
    Super Member
    Joined
    Jan 2009
    Posts
    715
    Quote Originally Posted by DavidEriksson View Post
    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 .
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