Section 9.6 Power Series
So far, our study of series has examined the question of “Is the sum of these infinite terms finite?,” i.e., “Does the series converge?” We now approach series from a different perspective: as a function. Given a value of we evaluate by finding the sum of a particular series that depends on (assuming the series converges). We start this new approach to series with a definition.
Example 9.6.3. Examples of power series.
Write out the first five terms of the following power series:
Solution 1.
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One of the conventions we adopt is that
regardless of the value of ThereforeThis is a geometric series in with -
This series is centered at
Note how this series starts with We could rewrite this series starting at with the understanding that and hence the first term is
Solution 2. Video solution
We introduced power series as a type of function, where a value of is given and the sum of a series is returned. Of course, not every series converges. For instance, in part 1 of Example 9.6.3, we recognized the series as a geometric series in Theorem 9.2.7 states that this series converges only when
This raises the question: “For what values of will a given power series converge?,” which leads us to a theorem and definition.
Theorem 9.6.4. Convergence of Power Series.
Let a power series be given. Then one of the following is true:
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The series converges only at
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The series converges for all
The value of is important when understanding a power series, hence it is given a name in the following definition. Also, note that part 2 of Theorem 9.6.4 makes a statement about the interval but the not the endpoints of that interval. A series may/may not converge at these endpoints.
Definition 9.6.6. Radius and Interval of Convergence.
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The number
given in Theorem 9.6.4 is the radius of convergence of a given series. When a series converges for only we say the radius of convergence is 0, i.e., When a series converges for all we say the series has an infinite radius of convergence, i.e., -
The interval of convergence is the set of all values of
for which the series converges.
To find the interval of convergence, we start by using the ratio test to find the radius of convergence If we know the series converges on and it remains to check for convergence at the endpoints.
Given we apply the ratio test to since the ratio test requires positive terms. We find
where It follows that the series converges absolutely (and therefore converges) for any such that that is, for in
On the other hand, suppose for some that Then, for sufficiently large This means that the terms of are growing in absolute value, and therefore cannot converge to zero. This means that the series diverges, by Theorem 9.2.23.
From the above observations, it follows that must be the radius of convergence.
Key Idea 9.6.7. Determining the Radius and Interval of Convergence.
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To determine the interval of convergence, plug the endpoints (
and ) into the power series, and test the resulting series for convergence. If the series converges, we include the endpoint. If it diverges, we exclude the endpoint.
Key Idea 9.6.7 allows us to find the radius of convergence of a series by applying the Ratio Test (or any applicable test) to the absolute value of the terms of the series. We practice this in the following example.
Example 9.6.8. Determining the radius and interval of convergence.
Find the radius and interval of convergence for each of the following series:
Solution 1.
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We apply the Ratio Test to the seriesThe Ratio Test shows us that regardless of the choice of
the series converges. Therefore the radius of convergence is and the interval of convergence is -
We apply the Ratio Test to the seriesThe Ratio Test states a series converges if the limit of
We found the limit above to be therefore, the power series converges when or when is in Thus the radius of convergence is To determine the interval of convergence, we need to check the endpoints of When we have the opposite of the Harmonic Series:The series diverges when When we have the series which is the Alternating Harmonic Series, which converges. Therefore the interval of convergence is -
We apply the Ratio Test to the seriesAccording to the Ratio Test, the series converges when
The series is centered at 3, and must be within of 3 in order for the series to converge. Therefore the radius of convergence is and we know that the series converges absolutely for all in We check for convergence at the endpoints to find the interval of convergence. When we have:which diverges. A similar process shows that the series also diverges at Therefore the interval of convergence is -
We apply the Ratio Test toThe Ratio Test shows that the series diverges for all
except Therefore the radius of convergence is
Solution 2. Video solution
We can use a power series to define a function:
where the domain of is a subset of the interval of convergence of the power series. One can apply calculus techniques to such functions; in particular, we can find derivatives and antiderivatives.
Theorem 9.6.9. Derivatives and Indefinite Integrals of Power Series Functions.
A few notes about Theorem 9.6.9:
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The theorem states that differentiation and integration do not change the radius of convergence. It does not state anything about the interval of convergence. They are not always the same.
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Notice how the summation for
starts with This is because the constant term of becomes through differentiation. -
Differentiation and integration are simply calculated term-by-term using the Power Rules.
Example 9.6.11. Derivatives and indefinite integrals of power series.
Solution 1.
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In Example 9.6.3, we recognized that
is a geometric series in We know that such a geometric series converges when that is, the interval of convergence is To determine the interval of convergence of we consider the endpoints ofTherefore, the interval of convergence of is -
To find the interval of convergence of we again consider the endpoints ofThe value of is irrelevant; notice that the rest of the series is an Alternating Series that whose terms converge to 0. By the Alternating Series Test, this series converges. (In fact, we can recognize that the terms of the series after are the opposite of the Alternating Harmonic Series. We can thus say that )Notice that this summation is the Harmonic Series, which diverges. Since converges for and diverges for the interval of convergence of is
Solution 2. Video solution
The previous example showed how to take the derivative and indefinite integral of a power series without motivation for why we care about such operations. We may care for the sheer mathematical enjoyment “that we can”, which is motivation enough for many. However, we would be remiss to not recognize that we can learn a great deal from taking derivatives and indefinite integrals.
Recall that in Example 9.6.11 is a geometric series. According to Theorem 9.2.7, this series converges to when Thus we can say
Integrating the power series, (as done in Example 9.6.11,) we find
while integrating the function gives
We established in Example 9.6.11 that the series on the left converges at substituting on both sides of the above equality gives
On the left we have the opposite of the Alternating Harmonic Series; on the right, we have We conclude that
Important: We stated in Key Idea 9.2.20 (in Section 9.2) that the Alternating Harmonic Series converges to and referred to this fact again in Example 9.5.6 of Section 9.5. However, we never gave an argument for why this was the case. The work above finally shows how we conclude that the Alternating Harmonic Series converges to
We use this type of analysis in the next example.
Example 9.6.12. Analyzing power series functions.
Solution 1.
We start by making two notes: first, in Example 9.6.8, we found the interval of convergence of this power series is Second, we will find it useful later to have a few terms of the series written out:
We now find the derivative:
Since the series starts at and each term refers to we can re-index the series starting with
We found the derivative of is The only functions for which this is true are of the form for some constant As (see Equation (9.6.3)), must be 1. Therefore we conclude that
for all
Solution 2. Video solution
Example 9.6.12 and the work following Example 9.6.11 established relationships between a power series function and “regular” functions that we have dealt with in the past. In general, given a power series function, it is difficult (if not impossible) to express the function in terms of elementary functions. We chose examples where things worked out nicely.
In this section’s last example, we show how to solve a simple differential equation with a power series.
Example 9.6.13. Solving a differential equation with a power series.
Solution.
The differential equation describes a function where the derivative of is twice and This is a rather simple differential equation; with a bit of thought one should realize that if then and hence By letting we satisfy the initial condition of
Let’s ignore the fact that we already know the solution and find a power series function that satisfies the equation. The solution we seek will have the form
In Section 9.8, as we study Taylor Series, we will learn how to recognize this series as describing
Our last example illustrates that it can be difficult to recognize an elementary function by its power series expansion. It is far easier to start with a known function, expressed in terms of elementary functions, and represent it as a power series function. One may wonder why we would bother doing so, as the latter function probably seems more complicated. In the next two sections, we show both how to do this and why such a process can be beneficial.
Exercises Exercises
Terms and Concepts
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What is the difference between the radius of convergence and the interval of convergence?
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Problems
Exercise Group.
In the following exercises, write out the sum of the first 5 terms of the given power series.
Exercise Group.
In the following exercises, a power series is given.
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Exercise Group.
In the following exercises, a function is given.
Exercise Group.
In the following exercises, give the first 5 terms of the series that is a solution to the given differential equation.
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