Integrating a power series (continued)

 In the previous post I showed how we can integrate a power series term by term to find the power representation of other functions. For example, we can integrate the power series of the function f(x) = ln(1+x) to find the power series representation of the f(x) = 1/1+x.

Example 

Solution

a) The derivative of f(x) = ln(1+x) is f'(x) = 1/1+x

Let's find a power series representation of f'(x) in order to integrate it:

The integral ∫f'(x)dx represents the antiderivatives of ln(1+x). We have:

ln(1+x) = C +  x  -x/2² +x³/3- x⁴/4 + ....

Since the power series representation of f'(x) is defined for ❙x❙<1 or -1<x<1, let's find the antiderivative for x = 0

ln( 1 + 0) = C + 0 + 0 + 0 + ....

ln1 = C lni1 = 0. Therefore C = 0.

Let's substitute C:

ln(1+x) =  x  -x²/2 +x³/3- x⁴/4 + 

 This represents an an alternating series that has the form:

The above expression is valid for ❘x❘<1. At x=1 we have by substituting x in the first expression of ln(x+1):

ln2 = 1-1/2 +1/3-1/4. We have an alternating harmonic series that converges.

At x =-1 we have ln0 = -1-1/2-1/3-1/4, which is an harmonic series that diverges. The interval of convergence is (-1,1)

Practice. Do the second exercise in the example

Differentiating and Integrating a power series

 Objectives:

1) Differentiate power series term by term

2) Integrate power series term by term

Let's f be the function that represents this power series. We can write:

The function f being a polynomial, we can differentiate and integrate it by differentiating and integrating each term of the polynomial in the same way we do it for a regular polynomial:

Differentiating f we have:

Integrating f we have:

Both the derivative and the integral of f converge on the interval I. Differentiating and integrating the function f this way is called differentiating and integrating term by term the power series that represents the function f .This property allows us to do two things. First, knowing the power series representation of a function f, we can find the power series representation of its derivative. Also knowing the power series representation of a function f allows us to find the power series representation of its integral. For example if I know the power series representation of  the function f(x) = 1/x-1, I can  differentiate term by term find the power series representation of g(x) = 1/(x-1)², which is the derivative of f . Similarly, knowing the power series representation of f(x) = 1/1+x, I can integrate term by term to find the power series representation g(x) = ln(1+x), which is the integral of f(x).

Theorem 

Example

Solution

Practice

Multiplication of power series

 Objective: Multiply 2 power series

Multiplying two power series create another power series. The multiplication of two power series allows to find power series representations for functions. The way we multiply power series is similar to the way we multiply polynomials. For example, suppose we want to multiply:

  

Example: Multiply the power series representation of the following function

Solution

Practice

Find the function represented by a power series

 Objective: Find the function represented by a power series

In the previous post, a function was given. The task was to find the corresponding power series. In this post we do the inverse problem. A power series is given. We need to find the corresponding function

Example

Solution

This is a geometric series that is convergent only and only if ❘2x❘<1 i.e -1<2x<1

-1/2<x<1/2. The series is convergent in the interval (-1/2, 1/2)

Practice

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

 

Properties of power series

 Properties of power series involve the following operations:

1) The addition and subtraction of power series

2) The multiplication of a power series by a constant or a power of the variable

3) The multiplication of two power series

4) The differentiation and integration of power series.

In the last post we represent certain functions in term of power series. The properties of power series help to facilitate this process. They allow us to find the power series representations of certain elementary functions by re-writing them in terms of functions of known series representations.

Combining power series

If we have two power series in the same interval of convergence, we can add or subtract these two power series to obtain a new series in the same interval of convergence. Similarly, we can multiply a power series by by a power of x that leads to a new power series. These properties allow us as we mentioned earlier to find the power series representation of certain functions knowing the power series representations of other functions . For example knowing the power series representation of f(x) = 1/1-x we can find the power representation of f(x) = 1/(1-x)² and y = 1/(x-1)(x-3).

Theorem

Example

Solution

for all x in the interval (-1,1).

According to the properties of combining power series above, the series

Practice