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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 $n$.

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

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    $\begingroup$ A neat answer by JM and some references. $\endgroup$ – Raymond Manzoni May 4 '16 at 22:34
  • $\begingroup$ @RaymondManzoni: splendid ! $\endgroup$ – Yves Daoust May 4 '16 at 22:38
  • $\begingroup$ Glad you liked it! Cheers, $\endgroup$ – Raymond Manzoni May 4 '16 at 22:43
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    $\begingroup$ @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 :) $\endgroup$ – Yves Daoust May 4 '16 at 22:48