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   Author  Topic: Floor Summation  (Read 8172 times)
ThudnBlunder
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Floor Summation  
« on: Nov 9th, 2009, 6:00am »
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Evaluate
       
81ntanh/10n
   n=1
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Obob
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Re: Floor Summation  
« Reply #1 on: Nov 9th, 2009, 2:18pm »
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It's roughly 1 - 2.413 * 10-264.  Are you looking for an actual precise answer, or is the point just that tanh pi is close to 1?
« Last Edit: Nov 9th, 2009, 2:50pm by Obob » IP Logged
ThudnBlunder
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Re: Floor Summation  
« Reply #2 on: Nov 11th, 2009, 3:25am »
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on Nov 9th, 2009, 2:18pm, Obob wrote:
It's roughly 1 - 2.413 * 10-264.  Are you looking for an actual precise answer, or is the point just that tanh pi is close to 1?

As I am not expecting an exact answer, perhaps I should have put this elsewhere.
The point is that the answer, a transcendental number, requires at least 239 decimal places before we can discover it does not equal 1. And even more if we want to consider rounding errors.
 
Do you normally sum series to such precision?  Smiley
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Obob
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Re: Floor Summation  
« Reply #3 on: Nov 11th, 2009, 6:10am »
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I just observed that the first time floor(n tanh pi) != n-1 occurs around n = 267 or so, summed the series 81(n-1)/10^n = 1, and then gave as a rough error estimate an approximation of the difference between the two series.
 
I guess I'm just saying that there is no reason to use tanh pi except for obfuscation; the real result lurking here is that
 
limx->1- sum (floor(n x))/10^n = 1, and that the convergence occurs very quickly.
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