The function

if different from the Exponential function or the Sine function.

We can calculate the Taylor's series at x=0 easily, using Newton's Binomial Theorem, but the function is not defined when x is smaller than -1.

"In the interval (-1, +1) the parabolas approach the original curve more and more as the order increases; but to the right of x = 1 they deviate from it increasingly, now above, now below, in a striking way." (Felix Klein)

At the singular point x = -1, the branch of the original curve which appears in the drawing ends at a vertical tangent, "all the parabolas extends beyond this point but approach the original curve more and more at x = -1, by becoming ever steeper". At the point x = +1, symmetrical to x = -1, the parabolas approach the original curve more and more closely.

LINKS

	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: 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.