structural:scholar_edition, information:highlighting, structural:reformatting
Asymptotic (divergent) series

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 E 1 (10) {\displaystyle E_{1}(10)}. However, for positive values of x, there is a divergent series approximation that can be obtained by integrating x e x E 1 (x) {\displaystyle xe^{x}E_{1}(x)} by parts:

The relative error of the approximation above is plotted on the figure to the right for various values of N {\displaystyle N}, the number of terms in the truncated sum (N = 1 {\displaystyle N=1} in red, N = 5 {\displaystyle N=5} in pink).