Creating Power Series From Functions

The Geometric Power Series

Recall that

Substituting x for r, we have

We write

Milking the Geometric Power Series

By using substitution, we can obtain power series expansions from the geometric series.

Example 1

Substituting x2 for x, we have

Example 2

Multiplying by x we have

Example 3

Suppose we want to find the power series for

1
f(x)  =
2x - 3

centered at x = 4.  We rewrite the function as

1                            1
=
2(x - 4) + 8 - 3          2(x - 4) + 5

Example 4

Substituting -x for x, we have

Example 5

Substituting x2 in for x in the previous example, we have

Example 6

Taking the integral of the previous example, we have

Exercise  Find the power series that represents the following functions:

1. ln(1 + x)

2. tanh-1x

3. -(1 - x)-2

Integrating Impossible Functions

We can use power series to integrate functions where there are no standard techniques of integration available.

Example:

Use power series to find the integral

Then use this integral to approximate

Solution:

Notice that this is a very difficult integral to solve.  We resort to power series.  First we use the series expansion from Example 6, replacing x with x2.

Integrating we arrive at the solution

Now to solve the definite integral, notice that when we plug in 0 we get 0, hence the definite integral is

Using the first 5 terms to approximate this we get 0.300

Notice that the error is less than the next term (which comes from x23/253)

E < 1/253  =  .004.