

Title: Interesting Limit Post by Barukh on Sep 2^{nd}, 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.gif_{k = 1...n} (n^{2} + k^{2})^{1} 

Title: Re: Interesting Limit Post by pex on Sep 2^{nd}, 2011, 4:20am Isn't that just [hide]the Riemann sum for the integral of (1+x^{2})^{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 2^{nd}, 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 2^{nd}, 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 3^{rd}, 2011, 2:01am ;D For the sake of completeness: [hideb]Multiply and divide by n^{2} to get lim_{n to inf} (1/n) sum_{k=1..n} (1 + (k/n)^{2})^{1}, which is by definition int_{0}^{1} (1 + x^{2})^{1} dx = arctan(1)  arctan(0) = pi/4.[/hideb] 

Powered by YaBB 1 Gold  SP 1.4! Forum software copyright © 20002004 Yet another Bulletin Board 