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Infinite series that converges to pi

  1. Feb 19, 2012 #1
    I stumbled upon this infinite series that converges to [itex]\pi[/itex]:

    4[itex]\sum\frac{\sqrt{n^2-i^2}}{n^2} [/itex] for i = 1:n as n[itex]{\rightarrow∞}[/itex]

    I haven't been able to find any similar series online and I'm really curious how to prove this does indeed converge to [itex]\pi[/itex]. Any insight would be greatly appreciated.
  2. jcsd
  3. Feb 19, 2012 #2


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  4. Feb 19, 2012 #3
    This series appear if you try to compute the area of a quarter unit circle, by approximating with n rectangles in the obvious way (one side being 1/n).

    EQuivalently, your sum is a Riemann sum of f(x)=√(1-x2) in the interval [0,1].
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