Combining Power series (continued)

 Objective:

 Construct the power series of functions using the power series representation of other functions  

 In this post I am going to find the power series of a given function  using the power series of known functions.  

Let's  start by finding the power series of a given function using the power series of known functions.

Example

Using the power series representation of f(x) = 1/1-x to find the power series representation of the following functions:

a) f(x) = 3x/1+x²

b) f(x) = 1/(x-1)(x-3)

Solution

Let's write f(x) under the form: f(x) = 1/1-x

f(x) = 3x(1/1-(-x²)

 The function between parenthesis represents the power series 

Therefore according to the second property of combining series the function f(x) can be represented by::

 

The interval of convergence of the new series is the set of real numbers such as ❘ x²❘<1 i.e -1<x<1

b) Let's decompose the function in partial functions:

1/(x-1)(x-3) = A/x-1 + B/x-3 

By reducing to the same denominator and factoring, we find:

1/(x-1)(x-3) = (A+B)x-3A-B/(x-1)(x-3)

By identification we find: 

A + B =0

-3A-B = 1

Resolving the system we find:

A =-1/2 and B = 1/2

Substituting A and B:

1/(x-1)(x-3) = -1/2/x--1 +1/2.1/x-3

                    = -1/2.1/x-1 + 1/2.1/x-3

                    = -1/2.1/-(1-x) +1/2.1/3./-(1-x/3)

                    = -1/2.-1/1-x+1/2.1/3/-(1-x/3)

                   = 1/2.1/1-x+1/6.1/1-x/3

The function is written under the form 

Such a series converges if x belongs to the interval (-1,1)

Practice