# Roots of the Taylor approximation of the exponential [duplicate]

While answering another question, I looked at the roots of the $n^{th}$ degree Taylor approximation of the exponential.

$$e^x\approx E_n(x):=\sum_{k=0}^n\frac{x^k}{k!}.$$

Apparently, these root are aligned on a parabola-like smooth curve depending on $n$.

How would you address the problem of finding the equation of this curve, ignoring the exact positions of the roots along it ?

• – Raymond Manzoni May 4 '16 at 22:34
• @RaymondManzoni: splendid ! – Yves Daoust May 4 '16 at 22:38
• Glad you liked it! Cheers, – Raymond Manzoni May 4 '16 at 22:43
• @RaymondManzoni: now, as these roots are the same as those of the incomplete Gamma $\Gamma(n+1,z)$, we even have a way to study a non-integer number of roots :) – Yves Daoust May 4 '16 at 22:48