Taylor polynomials: rational function 2
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. REFERENCES
Felix Klein  Elementary Mathematics from an Advanced Standpoint. Arithmetic, Algebra, Analysis (pags. 223228)  Dover Publications
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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 function is not defined for values less than 1. Taylor polynomials about the origin approximates the function between 1 and 1.
