Taylor's series

The Taylor's series is a power series, with the coefficients being the corresponding derivatives divided by the factorial of the order of the derivative:

y = yo + y'o (x - xo) + y''o (x - xo)^2 / 2 + y'''o (x - xo)^3 / 6 + ... + y[i]o (x - xo)^i / i! + ....

The Taylor series is applilcable to any function which has all derivatives.

Since the derivatives are unique, so is the Taylor's series. Hence, any method which obtains a power series for a given function actually obtains the Taylor's series. Thus, there are numerous methods of obtaining the Taylor's series. Usually at least one alternative method will be much easier than finding the derivatives.

The Taylor's series, being a power series, has a circle of convergence. Its radius extends to the nearest singularity. By analytic continuation (employing the translation), one may move the point about which the Taylor's series is exxpanded to any other point within the original circle of convergence. In this manner, one may walk the point about which the Taylor's series is expaned to anywhere that the function has all derivatives; provided that a suitable path can be found. That is, provided that the region of convergence is connected between the starting and finishing points.

The specific Taylor's series for the square root is displayed in the binomial.

Copywrite 1997 R. I. 'Scibor-Marchocki last modified on Sunday 03-rd of August 1997.