This rational function

is a continuos function. It has no singularities.

However, the behavior of the Taylor polynomial is no similar to other continuous functions such as the exponential function or the sinus function. Taylor series of these functions converge in all the real numbers.

Although the reason is not clear, the Taylor series is behaving in a manner similar to other function with two real singularities.

The Taylor series converges in an interval centered at the center of the development of Taylor but not outside it.

This behavior seems mysterious. "The mystery begins to unravel when we turn to the complex function

which is identical to H(x) when z is restricted to the real axis of the complex plane." (Tristan Needham)

It si clear that the function has two complex singularities at z = i and at z = -i. The Taylor series converges inside a circle with a radius of convergence that is the distance to the nearest singularity. And this is what we can see in the case of the real function: the radius of the interval of convergence is the distance from the center of the series to the point i in the complex plane.

LINKS

	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 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 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.
	Taylor polynomials: Exponential function By increasing the degree, Taylor polynomial approximates the exponential function more and more.
	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.