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 ready-made plots in [this <i>Mathematica</i> notebook](http://www.ma.utexas.edu/users/afenyes/share/truncated-exponential.nb), now available as a [PDF](http://www.ma.utexas.edu/users/afenyes/share/truncated-exponential.pdf)) 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 $e^x - 1$"?