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. 223-228) - 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.

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By increasing the degree, Taylor polynomial approximates the sine function more and more.