Taylor polynomials: Rational function 1
The rational function is, as in the example of the square root, 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 their ordinates are alternately equal to 1 and 0, while that of the original curve has the value 1/2. We can compare this behavior with that of the rational function 2. REFERENCES
Felix Klein  Elementary Mathematics from an Advanced Standpoint. Arithmetic, Algebra, Analysis (pags. 223228)  Dover Publications
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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.
