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        riddles >> putnam exam (pure math) >> Interesting Limit
        
(Message started by: Barukh on Sep 2^nd, 2011, 1:06am)

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} (n² + k²)^-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+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 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 n² to get lim_{n to inf} (1/n) sum_k=1..n (1 + (k/n)²)^-1, which is by definition int₀¹ (1 + x²)^-1 dx = arctan(1) - arctan(0) = pi/4.[/hideb]