Felix Klein started his chapter about Taylor's Theorem writing: "I shall depart, namely, from de usual treatment in the textbooks by bringing to the foreground the finite series, so important in practice, and by aiding the intuitive grasp of the situation by means of graphs. In this way it will all seem elementary and easily comprehensible."

These approximations can be made using polynomials of increasing degrees. They are easy to calculate and approximates the curve (parabolas of osculation). These Taylor polynomials are obtained by considering the first terms of Taylor's series.

We have to investigate whether and how far these polynomials represent usable curves of approximation.

In the case of the exponential function, as the order increases, the parabolas give usable approximation to the original curve for a greater and greater interval.

Taylor's series of the exponential function at x=0 is:

LINKS

	Taylor polynomials: Complex Exponential Function The complex exponential function is periodic. His power series converges everywhere in the complex plane.
	The Complex Exponential Function The Complex Exponential Function extends the Real Exponential Function to the complex plane.
	Mercator and Euler: Logarithm Function Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.
	Taylor polynomials: Sine function By increasing the degree, Taylor polynomial approximates the sine function more and more.
	Taylor polynomials: Square root The function is not defined for values less than -1. Taylor polynomials about the origin approximates the function between -1 and 1.
	Taylor polynomials: Rational function 1 The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.
	Taylor polynomials: Rational function 2 The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.