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 *Mathematica* 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 *k* is even, the truncation of degree $k$ has one nonzero real root. The location of this root depends *almost* 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 $e^x - 1$"?



Roots of truncations of $ e^x - 1$
I finally got around to googling this a bit, and I immediately came up with <http://www.mai.liu.se/~halun/complex/taylor/> which describes the same phenomenon for the exponential function itself. Briefly, if Pn is the Taylor polynomial of ex, then the zeroes of Pn(nx) pile up on the curve |ze1-z|=1, |z|≤1. They credit this discovery to Szegő (1924). See also the paper
*On the Zeros of the Taylor Polynomials Associated with the Exponential Function* by Brian Conrey and Amit Ghosh (The American Mathematical Monthly, **95**, No. 6 (1988), pp. 528-533), <http://www.jstor.org/stable/2322757> if you have JSTOR access.


