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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]



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