**Objective:**

Given *f(x)*, we want a power series expansion of this function
with respect to a chosen point *x _{o}*, as follows:

(1)

**Method:**

The general idea will be to process both sides of this equation and
choose values of *x* so that only one unknown appears each time.

To obtain *a _{o}*: Choose

To obtain *a _{1}*: First take the derivative of equation
(1)

(2)

Now choose
(3)

Now choose *x=x _{o}*.

To obtain *a _{3}*: First take the derivative of equation
(3)

(4)

Now choose *x=x _{o}*.

To obtain *a _{k}*: First take the

**Summary:**

The taylor series expansion of *f(x)* with respect to *x _{o}*
is given by:

**Generalization to multivariable
function:**

Let *x*, *y* and *z* be the three independent variables,

For the general case of *n* independent variables,

where the coefficients are given by,

(**Note:** the procedure above does not guarantee that the infinite
series converges. Please see Jenson and Jeffreys, *Mathematical Methods
in Chemical Engineering*, Academic Press, 1977, for a thorough discussion
on how to analyze the convergence of the resulting series.)

This page is maintained by Tomas B. Co (tbco@mtu.edu). Last revised 06/10/2007.

Tomas B. Co

Associate Professor

Department of Chemical Engineering

Michigan Technological University

1400 Townsend Avenue

Houghton, MI 49931-1295