Editing Exponential integral (section)

===Asymptotic (divergent) series===
[[Image:AsymptoticExpansionE1.png|right|200px|thumb| Relative error of the asymptotic approximation for different number <math>~N~</math> of terms in the truncated sum]]

Unfortunately, the convergence of the series above is slow for arguments of larger modulus. For example, more than 40 terms are required to get an answer correct to three significant figures for <math>E_1(10)</math>.<ref>Bleistein and Handelsman, p.&nbsp;2</ref> However, for positive values of x, there is a divergent series approximation that can be obtained by integrating <math>x e^x E_1(x)</math> by parts:<ref>Bleistein and Handelsman, p.&nbsp;3</ref>
: <math>E_1(x)=\frac{\exp(-x)} x \left(\sum_{n=0}^{N-1} \frac{n!}{(-x)^n} +O(N!x^{-N}) \right)</math>
The relative error of the approximation above is plotted on the figure to the right for various values of <math>N</math>, the number of terms in the truncated sum (<math>N=1</math> in red, <math>N=5</math> in pink).

==== Asymptotics beyond all orders ====
Using integration by parts, we can obtain an explicit formula<ref>{{Citation |last=O’Malley |first=Robert E. |title=Asymptotic Approximations |date=2014 |url=https://doi.org/10.1007/978-3-319-11924-3_2 |work=Historical Developments in Singular Perturbations |pages=27–51 |editor-last=O'Malley |editor-first=Robert E. |access-date=2023-05-04 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-11924-3_2 |isbn=978-3-319-11924-3|url-access=subscription }}</ref><math display="block">\operatorname{Ei}(z) = \frac{e^{z}} {z} \left (\sum _{k=0}^{n} \frac{k!} {z^{k}} + e_{n}(z)\right), \quad e_{n}(z) \equiv (n + 1)!\ ze^{-z}\int _{ -\infty }^{z} \frac{e^{t}} {t^{n+2}}\,dt</math>For any fixed <math>z</math>, the absolute value of the error term <math>|e_n(z)|</math> decreases, then increases. The minimum occurs at <math>n\sim |z|</math>, at which point <math>\vert e_{n}(z)\vert \leq \sqrt{\frac{2\pi } {\vert z\vert }}e^{-\vert z\vert }</math>. This bound is said to be "asymptotics beyond all orders".