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.