During a talk I was at today, the speaker mentioned that if you truncate the Taylor series for $e^x - 1$, you'll get lots of roots with nonzero real part, even though the full Taylor series only has pure imaginary roots. If you plot the roots of truncations of $e^x - 1$ (or check out the example plots below), you can see lots of cool features. I'd like to know where they come from! I know there's a vast literature on polynomials, but I'm a total beginner, and I don't know where to start. Here are a few specific questions: 1. The roots of a high-degree truncation seem to fall into two categories: roots that lie very close to the imaginary axis, and roots that lie on a C-shaped curve. (Another interpretation is that all of the roots lie on a curve, which has a very sharp kink near the imaginary axis.) Can you write down an equation for the curve? 2. If you put the roots of a lot of consecutive truncations together on the same plot, you'll see definite "stripes" to the right of the imaginary axis. Once a stripe appears, each higher-degree truncation sticks another root onto the end, making the stripe grow outward. Can you write down equations for the stripes? 3. If $k$ is odd, the truncation of degree $k$ has no nonzero real roots. If <i>k</i> is even, the truncation of degree $k$ has one nonzero real root. The location of this root depends <i>almost</i> linearly on $k$. Why is the dependence so close to linear? Does it get more linear as $k$ increases, or less? 4. Can roots be given identities that persist across time? That is, as $k$ increases, can you point to a sequence of roots and say, "those are all the same individual, which was born at $k$ = so-forth, is following such-and-such trajectory, and will grow up to become the root ($2\pi i\cdot$ whatever) of $e^x - 1$"? ### Example plots Each color in this plot represents the roots of a different truncation. The roots move out from the origin as the degree of the truncation grows. This plot only shows truncations at multiples of ten terms. [![Roots of truncations at multiples of ten terms][1]][1] The next two plots show all truncations out to sixty terms. [![Roots of all truncations, with multiples of ten terms highlighted][2]][2] [![Roots of all truncations][3]][3] The negative roots of a given truncation *almost* form a perfect semicircle around zero, but this plot reveals tiny differences in their absolute values (unless there's a systematic problem with *Mathematica*'s root estimation, which seems to be numerical for high-degree polynomials). [![The negative roots almost form a perfect semicircle][4]][4] If $k$ is odd, the truncation of degree $k$ has no nonzero real roots. If $k$ is even, the truncation of degree $k$ has one nonzero real root. The location of this root depends *almost* linearly on $k$, but the residual plot shows nonlinearity (again, unless there's a systematic problem with Mathematica's root estimation). [![Linear fit of real root vs. truncation degree][5]][5] [![Residual plot of real root vs. truncation degree][6]][6] [1]: https://i.sstatic.net/0B0bpyCY.png [2]: https://i.sstatic.net/zOrf1Jp5.png [3]: https://i.sstatic.net/2fwzwLBM.png [4]: https://i.sstatic.net/Kn5oggHG.png [5]: https://i.sstatic.net/f5ioIBR6.png [6]: https://i.sstatic.net/M61eYQGp.png