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Topic: Interesting Limit (Read 7568 times) 

Barukh
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Interesting Limit
« on: Sep 2^{nd}, 2011, 1:06am » 
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Find the limit of the following sum when n > : n _{k = 1...n} (n^{2} + k^{2})^{1}


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pex
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Re: Interesting Limit
« Reply #1 on: Sep 2^{nd}, 2011, 4:20am » 
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Isn't that just the Riemann sum for the integral of (1+x^{2})^{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 2^{nd}, 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 2^{nd}, 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


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pex
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Re: Interesting Limit
« Reply #4 on: Sep 3^{rd}, 2011, 2:01am » 
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For the sake of completeness: hidden:  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. 


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