Proof of Tayor's theorem for analytic functions
"Figure 1: The circle of convergence C in the complex w plane"
By Cauchy's integral formula
Adding and subtracting the value in the denominator, and rewriting, we have
We may expand the factor into a geometric series, provided that meaning that points of
and lie inside and points of lie on and that is a disc of radius called the circle of convergence of the Taylor's series. See Figure 1.
From Cauchy's integral formulas we recognize
The only thing that remains is to show that the remainder vanishes as .
We note that
Thus we may define
which yields the estimate
Here the constant factor follows from the Maximum Modulus Theorem.
Named for Brook Taylor (1685–1731), English mathematician.