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Let $z \in \mathbb C \setminus(-\infty,0)$. It is known that $$E_1(z) = \cfrac{e^{-z}}{z+\cfrac{1}{1+\cfrac{1}{z+\cfrac{2}{1+\cfrac{2}{z+\cfrac{3}{1+\cdots}}}}}}.$$ For example, see http://functions.wolfram.com/GammaBetaErf/ExpIntegralE/10/

Let $f_n(z)$ be the $n$th convergent (or "approximant") of the continued fraction above. Is there a good "a posteriori" upper bound (in the sense of William B. Jones and R. I. Snell, Truncation Error Bounds for Continued Fractions, SIAM Journal on Numerical Analysis 6 No. 2 (1969), pp. 210-221) for the truncation error $|E_1(z)-f_n(z)|$? In other words, if one writes $$|E_1(z)-f_n(z)| = K_n(z)|f_n(z)-f_{n-1}(z)|,$$ where $K_n(z)$ is real and positive, then can one provide a good upper bound for $K_n(z)$?

(Note: I'm happy to assume that the imaginary part of $z$ is as small positive as you like.)

For a related question, see the math.SE question Formula for the error of the nth convergent of the continued fraction representation of $E_1(x)$.

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