How can we reconcile the exponential function with the fundamental theorem of algebra?

Question

For the purposes of this question let's consider the exponential function to be an infinite polynomial (the limit as $n$ goes to infinity of an $n$th degree polynomial). It stands to reason then that we might expect an infinite polynomial to have infinitely many roots, and this does hold for many other infinite series like the sine and cosine.

Let us define: $$\exp_n(x) = \sum_{k=0}^n \frac{x^k}{k!}$$ so that we have $\exp(x) = \exp_{\infty}(x)$.

All $\exp_n(x)$ for finite $n$ are $n$th-degree polynomials with $n$-complex roots, so let us write: $$\exp_n(x) = \frac{1}{n!} \prod_{i=1}^n (x-r_i)$$

My question is this: as $n$ $\to$ $\infty$, what happens to these $r_i$? Numerically I have determined that only odd $n$ have real roots, and as we take the odd partial sums of the exponential function, the real root gets closer and closer to $-\infty$, which concurs with what I expect. But how do we account for the multiplicities? Do all complex roots of the partial sums go to $-\infty$, and if so, how can we show that (bonus points for a visualization if possible)?

As for the multiplicities, truncated exponentials $\exp_n(x)$ are irreducible, so they cannot have repeated roots. This MO post might also have something useful. — Sil, May 01 '23 at 19:44
@Sil There $\bf Q[X]$ are polynomials with rational coefficients? So they still may happen to be reducible over indicators with real coefficients? — SBF, May 01 '23 at 19:48
@Ilya It will be reducible over reals, but that does not matter, irreducibility over rationals is enough, see Irreducible polynomial over $\mathbb{Q}$ can not have repeated root in $\mathbb{C}$ — Sil, May 01 '23 at 19:51
Also see these, maybe they answer your question at least partially: Approximating roots of the truncated Taylor series of $\exp$ by values of the Lambert W function and What are the properties of the roots of the incomplete/finite exponential series? — Sil, May 01 '23 at 19:52
@Sil I've meant in the answer to your link. Anyways, I better ask there — SBF, May 01 '23 at 19:57
One trivial remark is that the series above converges locally uniformly over $\mathbb C$. Since $\exp$ does not have roots, the roots of $\exp_n$ must go to infinity in the complex plane as $n\to\infty$. How they behave is not clear of course, but for instance from the coefficients of the polynomials, you know that the sum of the roots of $\exp_n$ is $-n$, so the average of the roots is the constant $-1$. This suggests that the real part of some roots has to escape to $+\infty$ if the real part of some other roots escape to $-\infty$ (unless I made some mistake) — Lorenzo Pompili, May 01 '23 at 20:16
Have a look here and its picture in particular ; btw, I have taken this fascinating picture as my logo as you can see. — Jean Marie, May 01 '23 at 20:23
@LorenzoPompili so if some of the real parts of the roots go to $\infty$, does that mean that there are more solutions to exp in the "extended' complex plane than just $-\infty$? I notice that the map of the roots forms an incomplete circle, meaning that the roots can't be 'too close' to the positive real axis — O.S., May 03 '23 at 00:24

How can we reconcile the exponential function with the fundamental theorem of algebra?

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