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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.

Taylor polynomials: Rational function. Circle of convergence | matematicasVisuales

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.

Taylor polynomials: Rational function. Behavior at x = 1 | matematicasVisuales
Taylor polynomials: Rational function. Behavior at x = 1 | matematicasVisuales

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. 223-228) - Dover Publications

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Taylor polynomials (5): Rational function 2
The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.

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Taylor polynomials (3): Square root
The function is not defined for values less than -1. Taylor polynomials about the origin approximates the function between -1 and 1.

MORE LINKS

Taylor polynomials (6): Rational function with two real singularities
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.
Taylor polynomials (7): Rational function without real singularities
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 and Euler: Logarithm Function
Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.
Taylor polynomials: Rational function with two complex singularities
We will see how Taylor polynomials approximate the function inside its circle of convergence.
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By increasing the degree, Taylor polynomial approximates the exponential function more and more.
Taylor polynomials (2): Sine function
By increasing the degree, Taylor polynomial approximates the sine function more and more.
Polynomial functions and derivative (1): Linear functions
The derivative of a lineal function is a constant function.
Polynomial functions and derivative (2): Quadratic functions
The derivative of a quadratic function is a linear function, it is to say, a straight line.
Polynomial functions and derivative (3): Cubic functions
The derivative of a cubic function is a quadratic function, a parabola.
Polynomial functions and derivative (4): Lagrange polynomials (General polynomial functions)
Lagrange polynomials are polynomials that pases through n given points. We use Lagrange polynomials to explore a general polynomial function and its derivative.