Summation bug? (HP-49G, 50g)

[Thomas Klemm]

(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.


Cheers
Thomas