Power series was introduced in the previous section. This section continues the discussion on the power series and focuses on specific types of power series such as the Taylor, Maclaurin and binomial series.

Estimation of the Function e^x
Using Taylor Polynomial

Estimation of the Function sin(x)
Using Taylor Polynomial

Estimation of the Function cos(x)
Using Taylor Polynomial

An important theorem, which serves as the basis for the discussion of the Taylor series, is stated without the proof as follows:

If a function f(x) is represented by a power series with a radius of convergence of R > 0, the coefficients are given by

Substituting the coefficients back into the series yields

This series is referred to as the Taylor series of a function f(x) centered at c.

Maclaurin series is a special case of the Taylor series, which can be obtained by setting c = 0:

As mentioned in the previous section, power series is important because they can be used to approximate functions arise in the fields of mathematics, science and engineering. A function f(x) can be represented using the partial sum of the series T_n(x) and the remainder term R_n(x) as

     f(x) = T_n(x) + R_n(x)

The partial sum is given by

     

along with the remainder term R_n(x) given as:

     

where z is a number between x and c. Note that the partial sum is a polynomial, and it is also known as the n^th-degree Taylor polynomial of f(x) centered at c while the above R_n(x) form is referred to as Lagrange' form of the remainder term.

Some of the common functions, which can be represented using the Taylor or Maclaurin series, are summarized as follows:

Another common series is the binomial series, which has the following form:

     

where k is any real number, , and .

Note that the binomial series converges when .