What are the properties of the roots of the incomplete/finite exponential series?

Solution 1:

The zeros of the scaled functions $f_N(Nz)$ do converge to an airfoil-like curve. See an animation here.

See also these:

• Zeros of truncated Taylor series by Jonas (see the references at the end) (2013)

• Zeroes of the partial sums of the exponential function by Zemke (2009)

• On the zeroes of the nth partial sum of the exponential series by Zemyan (2005)

• The zeros of the partial sums of the exponential series by Walker (2003)

Solution 2:

One more picture; here I rescale the radial distances from the origin to their logarithm; the roots of the polynomials $f_{16},f_{32},f_{64},f_{128}$ are shown, the magenta line is that for $f_{16}$. I find it interesting, that the radial positions fit nicely together, see the straight lines from the origin (the roots do not exactly match with the lines but remarkably good)