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An animation on Question 3 on Exercise sheet 8...

The Taylor series around the origin for the function f(z)=1/(1+z2) has radius of convergence 1. Here, we study what that actually means:

We look at the sequence of partial sums of the Taylor series in question. Now, the Taylor series converges inside the unit disk to f, but it diverges outside. Observe what this means for the partial sums (here we look at all partial sums up to order 32 in the Taylor expansion) to converge/diverge on certain regions.

Note: You might have to downscale this animation (Firefox: Ctrl & -) and/or use the full screen mode (Internet Explorer and Firefox: F11) to watch it in full. You can restart it by reloading the page.

For the colouring, see p. 25 in the lecture notes.


 

 
   © 2008 by Bernd Sing • • last update: 16.03.2009 (03/16/2009)