Learning Objectives
- 6.1.1 Identify a power series and provide examples of them.
- 6.1.2 Determine the radius of convergence and interval of convergence of a power series.
- 6.1.3 Use a power series to represent a function.
A power series is a type of series with terms involving a variable. More specifically, if the variable is x, then all the terms of the series involve powers of x. As a result, a power series can be thought of as an infinite polynomial. Power series are used to represent common functions and also to define new functions. In this section we define power series and show how to determine when a power series converges and when it diverges. We also show how to represent certain functions using power series.
Form of a Power Series
A series of the form
where x is a variable and the coefficients cn are constants, is known as a power series. The series
is an example of a power series. Since this series is a geometric series with ratio we know that it converges if and diverges if
Definition
A series of the form
is a power series centered at A series of the form
is a power series centered at
To make this definition precise, we stipulate that and even when and respectively.
The series
and
are both power series centered at The series
is a power series centered at
Convergence of a Power Series
Since the terms in a power series involve a variable x, the series may converge for certain values of x and diverge for other values of x. For a power series centered at the value of the series at is given by Therefore, a power series always converges at its center. Some power series converge only at that value of x. Most power series, however, converge for more than one value of x. In that case, the power series either converges for all real numbers x or converges for all x in a finite interval. For example, the geometric series converges for all x in the interval but diverges for all x outside that interval. We now summarize these three possibilities for a general power series.
Theorem 6.1
Convergence of a Power Series
Consider the power series The series satisfies exactly one of the following properties:
- The series converges at and diverges for all
- The series converges for all real numbers x.
- There exists a real number such that the series converges if and diverges if At the values x where the series may converge or diverge.
Proof
Suppose that the power series is centered at (For a series centered at a value of a other than zero, the result follows by letting and considering the series We must first prove the following fact:
If there exists a real number such that converges, then the series converges absolutely for all x such that
Since converges, the nth term as Therefore, there exists an integer N such that for all Writing
we conclude that, for all
The series
is a geometric series that converges if Therefore, by the comparison test, we conclude that also converges for Since we can add a finite number of terms to a convergent series, we conclude that converges for
With this result, we can now prove the theorem. Consider the series
and let S be the set of real numbers for which the series converges. Suppose that the set Then the series falls under case i. Suppose that the set S is the set of all real numbers. Then the series falls under case ii. Suppose that and S is not the set of real numbers. Then there exists a real number such that the series does not converge. Thus, the series cannot converge for any x such that Therefore, the set S must be a bounded set, which means that it must have a smallest upper bound. (This fact follows from the Least Upper Bound Property for the real numbers, which is beyond the scope of this text and is covered in real analysis courses.) Call that smallest upper bound R. Since the number Therefore, the series converges for all x such that and the series falls into case iii.
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If a series falls into case iii. of Convergence of a Power Series, then the series converges for all x such that for some and diverges for all x such that The series may converge or diverge at the values x where The set of values x for which the series converges is known as the interval of convergence. Since the series diverges for all values x where the length of the interval is 2R, and therefore, the radius of the interval is R. The value R is called the radius of convergence. For example, since the series converges for all values x in the interval and diverges for all values x such that the interval of convergence of this series is Since the length of the interval is 2, the radius of convergence is 1.
Definition
Consider the power series The set of real numbers x where the series converges is the interval of convergence. If there exists a real number such that the series converges for and diverges for then R is the radius of convergence. If the series converges only at we say the radius of convergence is If the series converges for all real numbers x, we say the radius of convergence is (Figure 6.2).
To determine the interval of convergence for a power series, we typically apply the ratio test. In Example 6.1, we show the three different possibilities illustrated in Figure 6.2.
Example 6.1
Finding the Interval and Radius of Convergence
For each of the following series, find the interval and radius of convergence.
Solution
- To check for convergence, apply the ratio test. We have
for all values of x. Therefore, the series converges for all real numbers x. The interval of convergence is and the radius of convergence is - Apply the ratio test. For we see that
Therefore, the series diverges for all Since the series is centered at it must converge there, so the series converges only for The interval of convergence is the single value and the radius of convergence is - In order to apply the ratio test, consider
The ratio if Since implies that the series converges absolutely if The ratio if Therefore, the series diverges if or The ratio test is inconclusive if The ratio if and only if or We need to test these values of x separately. For the series is given by
Since this is the alternating harmonic series, it converges. Thus, the series converges at For the series is given by
This is the harmonic series, which is divergent. Therefore, the power series diverges at We conclude that the interval of convergence is and the radius of convergence is
Checkpoint 6.1
Find the interval and radius of convergence for the series
Representing Functions as Power Series
Being able to represent a function by an “infinite polynomial” is a powerful tool. Polynomial functions are the easiest functions to analyze, since they only involve the basic arithmetic operations of addition, subtraction, multiplication, and division. If we can represent a complicated function by an infinite polynomial, we can use the polynomial representation to differentiate or integrate it. In addition, we can use a truncated version of the polynomial expression to approximate values of the function. So, the question is, when can we represent a function by a power series?
Consider again the geometric series
Recall that the geometric series
converges if and only if In that case, it converges to Therefore, if the series in Example 6.3 converges to and we write
As a result, we are able to represent the function by the power series
We now show graphically how this series provides a representation for the function by comparing the graph of f with the graphs of several of the partial sums of this infinite series.
Example 6.2
Graphing a Function and Partial Sums of its Power Series
Sketch a graph of and the graphs of the corresponding partial sums for on the interval Comment on the approximation as N increases.
Solution
From the graph in Figure 6.3 you see that as N increases, becomes a better approximation for for x in the interval
Checkpoint 6.2
Sketch a graph of and the corresponding partial sums for on the interval
Next we consider functions involving an expression similar to the sum of a geometric series and show how to represent these functions using power series.
Example 6.3
Representing a Function with a Power Series
Use a power series to represent each of the following functions Find the interval of convergence.
Solution
- You should recognize this function f as the sum of a geometric series, because
Using the fact that, for is the sum of the geometric series
we see that, for
Since this series converges if and only if the interval of convergence is and we have
- This function is not in the exact form of a sum of a geometric series. However, with a little algebraic manipulation, we can relate f to a geometric series. By factoring 4 out of the two terms in the denominator, we obtain
Therefore, we have
The series converges as long as (note that when the series does not converge). Solving this inequality, we conclude that the interval of convergence is and
for
Checkpoint 6.3
Represent the function using a power series and find the interval of convergence.
In the remaining sections of this chapter, we will show ways of deriving power series representations for many other functions, and how we can make use of these representations to evaluate, differentiate, and integrate various functions.
Section 6.1 Exercises
In the following exercises, state whether each statement is true, or give an example to show that it is false.
converges at for any real numbers
If has radius of convergence and if for all n, then the radius of convergence of is greater than or equal to R.
Suppose that converges at At which of the following points might the series diverge? Use the fact that if converges at x, then it converges at any point closer to c than x.
Suppose that converges at At which of the following points must the series also converge? Use the fact that if converges at x, then it converges at any point closer to c than x.
In the following exercises, suppose that as Find the radius of convergence for each series.
In the following exercises, find the radius of convergence R and interval of convergence for with the given coefficients
In the following exercises, find the radius of convergence of each series.
In the following exercises, use the ratio test to determine the radius of convergence of each series.
In the following exercises, given that with convergence in find the power series for each function with the given center a, and identify its interval of convergence.
Use the next exercise to find the radius of convergence of the given series in the subsequent exercises.
Suppose that such that if n is even. Explain why
Suppose that converges on Find the interval of convergence of
In the following exercises, suppose that satisfies where for each n. State whether each series converges on the full interval or if there is not enough information to draw a conclusion. Use the comparison test when appropriate.
Suppose that is a polynomial of degree N. Find the radius and interval of convergence of
[T] Plot the graphs of and of the partial sums for on the interval Comment on the approximation of by near and near as N increases.
[T] Plot the graphs of and of the partial sums for on the interval Comment on the behavior of the sums near and near as N increases.
[T] Plot the graphs of the partial sums for on the interval Comment on the behavior of the sums near and near as N increases.
[T] Plot the graphs of the partial sums for on the interval Comment on the behavior of the sums near and near as N increases.
[T] Plot the graphs of the partial sums for on the interval Comment on how these plots approximate as N increases.
[T] Plot the graphs of the partial sums for on the interval Comment on how these plots approximate as N increases.