As in the case of the Exponential function, the parabolas give usable approximations to the original curve for a greater and greater interval. The parabolas make the effort to share more and more oscillations with the sine curve. (Felix Klein)

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

LINKS

	The Complex Cosine Function The Complex Cosine Function extends the Real Cosine Function to the complex plane. It is a periodic function that shares several properties with his real ancestor.
	The Complex Cosine Function: mapping an horizontal line The Complex Cosine Function maps horizontal lines to confocal ellipses.
	Taylor polynomials: Exponential function By increasing the degree, Taylor polynomial approximates the exponential function more and more.
	Taylor polynomials: Complex Cosine Function The power series of the Cosine Function converges everywhere in the complex plane.
	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 with two real singularities This function has two real singularities at -1 and 1. Taylor polynomials approximate the function in an interval centered at the center of the series. Its radius is the distance to the nearest singularity.
	Taylor polynomials: Rational function without real singularities This is a continuos function and has no real singularities. However, the Taylor series approximates the function only in an interval. To understand this behavior we should consider a complex function.
	Taylor polynomials: Rational function with two complex singularities We will see how Taylor polynomials approximate the function inside its circle of convergence.
	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.