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)

  

 

  

The fundamentals of power series

 Learning objectives:

1. Identify a power series and provide examples of them

2. Determine the radius of convergence and the interval of convergence of a series

Definition  

A power series is a series with power of a variable. If the variable is x, the series holds power of x. Consequently a power series is an infinite polynomial. It is used to represent functions. A power series has the form:

where x is a variable and the coefficients cₙ are constants.

Example: The power series

is an example of a power series since it holds the powers of the variable x. It is a geometric series with ratio x. We know that it converges if ❘x❘<1 and diverges if ❘x❘>1.

Power series centered at x =a

A power series in the form

is a power series centered at x = a.

If a = 0, the power series is centered at 0. It's written as follow:

We stipulate that x⁰ = 1 and (x-a)⁰ = 1 even when x = 0 and x = a.

Examples:

1. The following series is centered at x = 2

2. 

are centered at x = 0.

Convergence of a power series 

One of the following is true:

1. The series converges at its center

2. It converges for all real numbers x

3. It converges in an interval

Theorem

Interval and radius of convergence. Definition

Graph of convergence of a power series

We apply the ratio test to determine the ratio test for a power series.

Examples

Solution

Note. It's better to do a discussion before concluding for the value of ρ according to the values of x. This value depends on a limit and an absolute value.

If x = 0, ρ = 0.1/∞ +1 =0

If x ≠ 0 ρ = ❘x❘.1/∞+1 = ❘x❘.0 =0. Therefore  ρ = 0<1 for all values of x  The series converges for all values of x

b. Let's apply the convergence test by calculating ρ :

Let's do a discussion here.

If x = 0, ρ = 0 ✕∞ (indetermination)

    If  x ≠ 0,  ρ = ∞. Therefore the series diverges for all values of x different of 0. Since it is centered at x = 0, it must converge at x = 0. Therefore the series convergences at the single point x = 0 and its radius of convergence is 0.

Practice

Find the interval and radius of convergence of

Strategies for testing the convergence or divergence of an infinite series

 Studying infinite series comes down to the study of their convergence or divergence. Unfortunately in this section of mathematics there is no definite method to determine the convergence or divergence of a series. In this post I show the strategies used for that purpose.

Many series fall into several types. Recognizing these types will help to know which test or strategies to use to determine the convergence or divergence a series.

Some series like the p-series and the geometric series are easy to determine their convergence.   

P-series

Comparison and limit comparison test

If a series is similar to either one of the series above, you can use the comparison or the limit comparison test.

Some series will not converge obviously. You will save a lot of work by recognizing this.

Divergence test 

The following two tests prove convergence but also the stronger fact that Σ❘aₙ| converges but also converges absolutely.

The above example contains powers and factorials. The ratio test is useful when aₙ contains powers and factorials

Using convergence steps. Examples

Example 1

Solution

We take the following steps:

1. The series is not an harmonic series, an alternating harmonic series a geometric or p-series

2. It is not an alternating series

3. It is not similar to a p-series or a geometric but we can approximate it to an harmonic series:

For larger n the series can be approximated as follows:

The series is similar to an harmonic series. Therefore we can use the comparison test or the limit comparison test. Using the limit comparison test we have:

The limit is a finite number. Since the harmonic series diverges, the given series diverges also.

Example 2

Solution

We take the following steps:

1. The series is not a familiar series: harmonic, p-series, geometric series

2. It is not an alternating series

3. There is no known series to compare it to

4. There is no factorial. 

5. There is a power but we cannot use the root test

6. Having used all the options for convergence or divergence let's use the divergence test

We have:

The series diverges according to the divergence test.

Practice

Determine whether the following series converges or diverges using the appropriate steps.