This Demonstration investigates how the roots of the polynomials in the Taylor expansion of a function analytic in the unit disk converge to the roots of the function in the disk. The roots of the selected analytic function are represented by small red points. The roots of the Taylor polynomial of a chosen degree are represented by larger green points. As you increase the degree of the Taylor polynomial, the number of green points may increase or decrease but eventually there will be as many red points as green points in the disk and they will almost coincide.
The fact that the roots of the polynomials in the Taylor expansion of an analytic function in the unit disk converge to the roots of the function of the disk follows from Hurwitz's theorem in complex analysis ( p. 119). More precisely: let be a function analytic in the unit disk and let be a sequence of analytic functions converging uniformly to on the disk. Then Hurwitz's theorem states that every point in the interior of a disk is contained in a smaller disk such that the number of roots of and (counted with multiplicity) in this smaller disk is the same for all large enough . From the compactness of the disk it follows that for sufficiently large , and will have the same number of roots in the unit disk and the sets of roots can be made arbitrarily close to each other.
 E. C. Titchmarch, The Theory of Functions, London: Oxford University Press, 1952.