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Summation bug? (HP-49G, 50g)
01-15-2015, 09:37 AM
Post: #8
RE: Summation bug? (HP-49G, 50g)
(01-15-2015 01:28 AM)Thomas Klemm Wrote:  
(01-13-2015 10:13 PM)Gerson W. Barbosa Wrote:  I have no idea why this works, however. Apparently the first four terms are correct. The last term was chosen somewhat arbitrarily to adjust the results to their correct 12 digits.

This is the Taylor-Series of \(\arccos(sinc(x))\):
\frac{1}{\sqrt{3}}(x-\frac{x^3}{90}-\frac{x^5}{4200}-\frac{x^7}{378000}+\frac{533 x^9}{6286896000}+\frac{2599 x^{11}}{454053600000}+\frac{75799 x^{13}}{572107536000000})+O(x^{14})

The correct value is \(\frac{6,286,896,000}{533}=\frac{17,325\cdot9!}{533}\approx11,795,302.0638\) instead of \(32\cdot9! = 11,612,160\).
Thus instead of 32 the factor is \(\frac{17,325}{533}\approx32.5047\).

Quote:This is a remarkable approximation of the sinc function
Now I hope it's clear why.

It is crystal clear now!

Thank you very much!

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RE: Summation bug? (HP-49G, 50g) - Gerson W. Barbosa - 01-15-2015 09:37 AM

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