The rational function
is, as in the example of the square root or in the rational function 1, 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 the ordinates increase indefinitely with the order, and alternate in sign.
We can compare this behavior with that of the rational function 1.
Klein wrote: "The theoretical consideration of Taylor's series cannot be made complete without going over to the complex variable.
It is only then that one can understand the sudden ceasing of the power series to converge at places where the function is entirely
regular. To be sure, one might be satisfied, in the case of our examples, by saying that the series cannot converge any farther
to the right than to the left, and that the convergence must cease at the left because of the singularity at x = -1." (Klein, p. 227)
We can see more examples to understand better this question, for example in
Taylor polynomials (6): Rational function with two real singularities.
Felix Klein - Elementary Mathematics from an Advanced Standpoint. Arithmetic, Algebra, Analysis (pags. 223-228) - Dover Publications
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.
Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.
We will see how Taylor polynomials approximate the function inside its circle of convergence.
By increasing the degree, Taylor polynomial approximates the exponential function more and more.
By increasing the degree, Taylor polynomial approximates the sine function more and more.
The function is not defined for values less than -1. Taylor polynomials about the origin approximates the function between -1 and 1.
The derivative of a lineal function is a constant function.
The derivative of a quadratic function is a linear function, it is to say, a straight line.
The derivative of a cubic function is a quadratic function, a parabola.
Lagrange polynomials are polynomials that pases through n given points. We use Lagrange polynomials to explore a general polynomial function and its derivative.