Representing a function as a power series

 If a power series converges to an expression representing a function, we can use the power series to represent the function. The function is then represented by an infinite polynomial. The polynomial expression can be used to differentiate or integrate the function. A truncated version of the polynomial expression can be used to approximate the values of the function. Let see how that can happen.

Let's consider the geometric series:

Let's recall that the following geometric series

converges if and only if ❘r❘<1. Therefore the series converges to a/1-a. 

Similarly the power series in x above converges if ❘x❘<1. It converges to 1/1-r. We can write:

In the following example we will show the power series provides a representation of the function f(x) = 1/1-x by comparing the graph of f with the graph of several of the partial sums of the power series.

Example 1

Solution

Let's graph f(x), S₂, S₄(x), S₆(x) in the interval (-1,1):

The graph shows a function and three approximations of it by partial sums of a power series. As N increases we get a better approximation of the function f

Example 2.

Solution

a) Let's recall that when ❘r❘<1 the series 

converges to a/1-r. Therefore a/1-r is the sum of the geometric series.

a/1-r = a + ar + ar² + .... (1)

Let's find a and r by writing the function f under the form a/1-r:

1/1+x³ = 1/1-(-x³)

We have a = 1, r = -x³. By substituting in (1), we get:

1/1+x³ = 1 + 1(-x³) + 1(-x³)² + ....= 1- x³ + x⁶ - ...

Since the series converges when ❘-x³❘<1, the interval of convergence is (-1,1)

Practice

Do example b)