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   Author  Topic: Interesting Limit  (Read 7568 times)
Barukh
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Interesting Limit  
« on: Sep 2nd, 2011, 1:06am »
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Find the limit of the following sum when n -> :
 
n k = 1...n (n2 + k2)-1
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pex
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Re: Interesting Limit  
« Reply #1 on: Sep 2nd, 2011, 4:20am »
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Isn't that just the Riemann sum for the integral of (1+x2)-1 over 0..1? That would make the limit equal to pi divided by four.
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Grimbal
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Re: Interesting Limit  
« Reply #2 on: Sep 2nd, 2011, 5:07am »
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Here is as formal as a proof as I could get in the short time I worked on this:
hidden:

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.

QED.
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Barukh
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Re: Interesting Limit  
« Reply #3 on: Sep 2nd, 2011, 11:40am »
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pex, you are right, and you probably know a  much more elegant proof than that of Grimbal's
 
 Grin
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pex
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Re: Interesting Limit  
« Reply #4 on: Sep 3rd, 2011, 2:01am »
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Grin For the sake of completeness:
 
hidden:
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.
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