``` wu :: forums (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi) riddles >> putnam exam (pure math) >> Interesting Limit (Message started by: Barukh on Sep 2nd, 2011, 1:06am) ``` Title: Interesting Limit Post by Barukh on Sep 2nd, 2011, 1:06am Find the limit of the following sum when n -> http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif:n http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/csigma.gifk = 1...n (n2 + k2)-1 Title: Re: Interesting Limit Post by pex on Sep 2nd, 2011, 4:20am Isn't that just [hide]the Riemann sum for the integral of (1+x2)-1 over 0..1[/hide]? That would make the limit [hide]equal to pi divided by four[/hide]. Title: Re: Interesting Limit Post by Grimbal on Sep 2nd, 2011, 5:07am Here is as formal as a proof as I could get in the short time I worked on this:[hideb]I computed the sum for n=1000.  I got 0.7866.  pi/4 = 0.7854.Between an extraordinary coincidence and a very plausible pex being correct, the second option is much more probable.[/hideb]QED. Title: Re: Interesting Limit Post by Barukh on Sep 2nd, 2011, 11:40am pex, [hide]you are right, and you probably know a  much more elegant proof than that of Grimbal's[/hide] ;D Title: Re: Interesting Limit Post by pex on Sep 3rd, 2011, 2:01am ;D For the sake of completeness:[hideb]Multiply and divide by n2 to get limn to inf (1/n) sumk=1..n (1 + (k/n)2)-1, which is by definition int01 (1 + x2)-1 dx = arctan(1) - arctan(0) = pi/4.[/hideb] Powered by YaBB 1 Gold - SP 1.4! Forum software copyright © 2000-2004 Yet another Bulletin Board