Taylor polynomials: Square root
The function if different from the Exponential function or the Sine function because this function is not defined for all values (its domain is not the whole real line), then it has no sense to say that a polynomial approaches the function for all values of x or that the power series converges for all values of x. 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. Then our interest is to study what happens when x is greater or equal to 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) We call this interval (1, +1) 'circle of convergence' of the power series. It is a typical behavior that a power series converges inside the circle of convergence and diverges outside. We understand better the use of the word 'circle' when we go to the Complex Plane. It is interesting to study what happens at the 'border' points. At the 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 too. We can see what happens with some functions with singularities, for example, Taylor polynomials (4): Rational function 1."
REFERENCES
Felix Klein  Elementary Mathematics from an Advanced Standpoint. Arithmetic, Algebra, Analysis (pags. 223228)  Dover Publications
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The function has a singularity at 1. Taylor polynomials about the origin approximates the function between 1 and 1.
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.
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.
The usual definition of a function is restrictive. We may broaden the definition of a function to allow f(z) to have many differente values for a single value of z. In this case f is called a manyvalued function or a multifunction.
