Taylor series

The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, who published the special case of the Taylor result in the mid 1700s.

and so the power series expansion agrees with the Taylor series. Thus a function is analytic in an open disk centred at b if and only if its Taylor series converges to the value of the function at each point of the disk.

The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin.

The usual trigonometric functions and their inverses have the following Maclaurin series:

The hyperbolic functions have Maclaurin series closely related to the series for the corresponding trigonometric functions:

And the formulas presented below are called inverse tangent integrals:

The complete elliptic Integrals of first kind K and of second kind E can be defined as follows:

The Jacobi theta functions describe the world of the elliptic modular functions and they have these Taylor series:

The regular partition number sequence P(n) has this generating function:

The strict partition number sequence Q(n) has that generating function:

In order to compute the 7th degree Maclaurin polynomial for the function

The Taylor series for the natural logarithm is (using the big O notation)

Then multiplication with the denominator and substitution of the series of the cosine yields

A second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as

which is to be understood as a still more abbreviated multi-index version of the first equation of this paragraph, with a full analogy to the single variable case.

Evaluating these derivatives at the origin gives the Taylor coefficients