Summing divergent asymptotic series

Question

I found the sine integral si to be

$$Si (x)\sim \frac \pi 2+\sum _{n=1}^\infty (-1)^n \left(\frac{(2 n-1)! \sin (x)}{x^{2 n}}+\frac{(2 n-2)! \cos (x)}{x^{2 n-1}}\right)$$

Say I want to find $Si(\frac \pi 4)$ what options have I got to use this divergent series to find the actual value?

Answer 1

In order to find a good approximation of $\text{Si}\left(\frac{\pi}{4}\right)$, I strongly suggest you to use a converging series and not a diverging one. For instance, the almost trivial: $$\text{Si}\left(\frac{\pi}{4}\right)=\int_{0}^{\pi/4}\sum_{n\geq 0}\frac{(-1)^n}{(2n+1)!}x^{2n}\,dx = \sum_{n\geq 0}\frac{(-1)^n \pi^{2n+1}}{4^{2n+1}(2n+1)(2n+1)!}$$ converges pretty fast.

Answer 2

For the computation of $\text{Si}(x)$, you can avoid summations and use Pade approximants $[n+1,n]$ and they are very accurate for $0<x<\pi$.

The best formulation write $$\text{Si}(x)=x \frac{P_n}{Q_n}$$

For example, for $n=8$ $$P_8=1-\frac{4094517636881845481 x^2}{104725905364944197604}+\frac{3080478831447814421 x^4}{3490863512164806586800}-$$ $$\frac{1307662727118635527267 x^6}{173447044465420580071744800}+\frac{8312356513454527739283241 x^8}{339990896561117421056634156960000}$$ $$Q_8=1+\frac{191509801858462833 x^2}{11636211707216021956}+\frac{151401002938061331 x^4}{1163621170721602195600}+$$ $$\frac{142463152299901321 x^6}{235982373422340925267680}+\frac{109740904565879261 x^8}{76143645824275338553038080}$$

Applied to $x=\frac \pi 4$, this approximation gives $$\text{Si}(\frac \pi 4)\approx 0.758975881068782699682925759337$$ while the "exact" value is $$\text{Si}(\frac \pi 4)\approx 0.758975881068782699681242985677$$

According to Wikipedia, the formula obtained fo $n=12$ is accurate to better than $10^{−16}$ for $0 ≤ x ≤ 4$.

While $$s_p=\sum_{n=0}^p\frac{(-1)^n \pi^{2n+1}}{4^{2n+1}(2n+1)(2n+1)!}$$ would give $$s_1=0.758482992668021382380004322324$$ $$s_2=0.758981071582059926412007482008$$ $$s_3=0.758975846410033901090466538410$$ $$s_4=0.758975881227999349736261215637$$ $$s_5=0.758975881068249561241406575169$$ $$s_6=0.758975881068784057647090162047$$ $$s_7=0.758975881068782696962863719284$$ $$s_8=0.758975881068782699685631148010$$ $$s_9=0.758975881068782699681237153361$$