The Universe of Discourse

Sat, 14 Sep 2019

Convergence of Taylor series

[ (Previously) ]

Wikipedia has an animation that shows the convergence of the Maclaurin series for the exponential function:

graphic, which focuses on x between -3 and +3, and y between -5 and
+20.  It displays the curve
y=exp(x) overlaid with its sequence of Maclaurin series
approximations, one after the other.

This makes it look a lot better-behaved than it is. First, because it only shows a narrow region around the center of convergence. But also, it emphasizes the right-hand quadrant, where the convergence is monotone.

If you focus on the left-hand side instead, you see the appoximations thrashing back and forth like a screen door in a hurricane:

The same,
but it focuses on x between -6 and 0, and y between -5 and 1.

This is the same as Wikipedia's animation, just focused on a slightly different region of the plane. But how different it looks!

(Individual frames are available)

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