The rational function

is as in the example of the square root, a special case of Newton's Binomial Theorem. We can calculate its Taylor's series at x = 0 easily.

This function has a singularity at the point x =- 1. The approximation is good in ranges from -1 to +1. Again we find a useful approach that is centered on the origin.

At the point x = +1 their ordinates are alternately equal to 1 and 0, while that of the original curve has the value 1/2. We can compare this behavior with that of the rational function 2.

LINKS

	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: 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: 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.