Create a new power series by multiplication by a power of the variable or a constant, or by substitution
Multiply two power series together
Combining Power Series
If we have two power series with the same interval of convergence, we can add or subtract the two series to create a new power series, also with the same interval of convergence. Similarly, we can multiply a power series by a power of x or evaluate a power series at xmxm for a positive integer m to create a new power series. Being able to do this allows us to find power series representations for certain functions by using power series representations of other functions. For example, since we know the power series representation for f(x)=11−xf(x)=11−x, we can find power series representations for related functions, such as
y=3x1−x2 and y=1(x−1)(x−3)y=3x1−x2 and y=1(x−1)(x−3).
In Combining Power Series we state results regarding addition or subtraction of power series, composition of a power series, and multiplication of a power series by a power of the variable. For simplicity, we state the theorem for power series centered at x=0x=0. Similar results hold for power series centered at x=ax=a.
theorem: Combining Power Series
Suppose that the two power series ∞∑n=0cnxn∞∑n=0cnxn and ∞∑n=0dnxn∞∑n=0dnxn converge to the functions f and g, respectively, on a common interval I.
The power series ∞∑n=0(cnxn±dnxn)∞∑n=0(cnxn±dnxn) converges to f±gf±g on I.
For any integer m≥0m≥0 and any real number b, the power series ∞∑n=0bxmcnxn∞∑n=0bxmcnxn converges to bxmf(x)bxmf(x) on I.
For any integer m≥0m≥0 and any real number b, the series ∞∑n=0cn(bxm)n∞∑n=0cn(bxm)n converges to f(bxm)f(bxm) for all x such that bxmbxm is in I.
Proof
We prove i. in the case of the series ∞∑n=0(cnxn+dnxn)∞∑n=0(cnxn+dnxn). Suppose that ∞∑n=0cnxn∞∑n=0cnxn and ∞∑n=0dnxn∞∑n=0dnxn converge to the functions f and g, respectively, on the interval I. Let x be a point in I and let SN(x)SN(x) and TN(x)TN(x) denote the Nth partial sums of the series ∞∑n=0cnxn∞∑n=0cnxn and ∞∑n=0dnxn∞∑n=0dnxn, respectively. Then the sequence {SN(x)}{SN(x)} converges to f(x)f(x) and the sequence {TN(x)}{TN(x)} converges to g(x)g(x). Furthermore, the Nth partial sum of ∞∑n=0(cnxn+dnxn)∞∑n=0(cnxn+dnxn) is
we conclude that the series ∞∑n=0(cnxn+dnxn)∞∑n=0(cnxn+dnxn) converges to f(x)+g(x)f(x)+g(x).
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We examine products of power series in a later theorem. First, we show several applications of Combining Power Series and how to find the interval of convergence of a power series given the interval of convergence of a related power series.
Example: Combining Power Series
Suppose that ∞∑n=0anxn∞∑n=0anxn is a power series whose interval of convergence is (−1,1)(−1,1), and suppose that ∞∑n=0bnxn∞∑n=0bnxn is a power series whose interval of convergence is (−2,2)(−2,2).
Find the interval of convergence of the series ∞∑n=0(anxn+bnxn)∞∑n=0(anxn+bnxn).
Find the interval of convergence of the series ∞∑n=0an3nxn∞∑n=0an3nxn.
Show Solution
Since the interval (−1,1)(−1,1) is a common interval of convergence of the series ∞∑n=0anxn∞∑n=0anxn and ∞∑n=0bnxn∞∑n=0bnxn, the interval of convergence of the series ∞∑n=0(anxn+bnxn)∞∑n=0(anxn+bnxn) is (−1,1)(−1,1).
Since ∞∑n=0anxn∞∑n=0anxn is a power series centered at zero with radius of convergence 1, it converges for all x in the interval (−1,1)(−1,1). By Combining Power Series, the series
∞∑n=0an3nxn=∞∑n=0an(3x)n∞∑n=0an3nxn=∞∑n=0an(3x)n
converges if 3x is in the interval (−1,1)(−1,1). Therefore, the series converges for all x in the interval (−13,13)(−13,13).
Watch the following video to see the worked solution to Example: Combining Power Series.
Suppose that ∞∑n=0anxn∞∑n=0anxn has an interval of convergence of (−1,1)(−1,1). Find the interval of convergence of ∞∑n=0an(x2)n.
Hint
Find the values of x such that x2 is in the interval (−1,1).
Show Solution
Interval of convergence is (−2,2).
In the next example, we show how to use Combining Power Series and the power series for a function f to construct power series for functions related to f. Specifically, we consider functions related to the function f(x)=11−x and we use the fact that
11−x=∞∑n=0xn=1+x+x2+x3+⋯
for |x|<1.
Example: Constructing Power Series from Known Power Series
Use the power series representation for f(x)=11−x combined with Combining Power Series to construct a power series for each of the following functions. Find the interval of convergence of the power series.
f(x)=3x1+x2
f(x)=1(x−1)(x−3)
Show Solution
First write f(x) as
f(x)=3x(11−(-x2)).
Using the power series representation for f(x)=11−x and parts ii. and iii. of Combining Power Series, we find that a power series representation for f is given by
∞∑n=03x(-x2)n=∞∑n=03(−1)nx2n+1.
Since the interval of convergence of the series for 11−x is (−1,1), the interval of convergence for this new series is the set of real numbers x such that |x2|<1. Therefore, the interval of convergence is (−1,1).
To find the power series representation, use partial fractions to write f(x)=1(1−x)(x−3) as the sum of two fractions. We have
First, using part ii. of Combining Power Series, we obtain
121−x=∞∑n=012xnfor|x|<1.
Then, using parts ii. and iii. of Combining Power Series, we have
161−x3=∞∑n=016(x3)nfor|x|<3.
Since we are combining these two power series, the interval of convergence of the difference must be the smaller of these two intervals. Using this fact and part i. of Combining Power Series, we have
1(x−1)(x−3)=∞∑n=0(12−16⋅3n)xn
where the interval of convergence is (−1,1).
try it
Use the series for f(x)=11−x on |x|<1 to construct a series for 1(1−x)(x−2). Determine the interval of convergence.
Hint
Use partial fractions to rewrite 1(1−x)(x−2) as the difference of two fractions.
Show Solution
∞∑n=0(−1+12n+1)xn. The interval of convergence is (−1,1).
In the previous example, we showed how to find power series for certain functions. In the next example we show how to do the opposite: given a power series, determine which function it represents.
Example: Finding the Function Represented by a Given Power Series
Consider the power series ∞∑n=02nxn. Find the function f represented by this series. Determine the interval of convergence of the series.
Show Solution
Writing the given series as
∞∑n=02nxn=∞∑n=0(2x)n,
we can recognize this series as the power series for
f(x)=11−2x.
Since this is a geometric series, the series converges if and only if |2x|<1. Therefore, the interval of convergence is (−12,12).
try it
Find the function represented by the power series ∞∑n=013nxn. Determine its interval of convergence.
Hint
Write 13nxn=(x3)n.
Show Solution
f(x)=33−x. The interval of convergence is (−3,3).
Try It
Multiplication of Power Series
We can also create new power series by multiplying power series. Being able to multiply two power series provides another way of finding power series representations for functions.
The way we multiply them is similar to how we multiply polynomials. For example, suppose we want to multiply
In Multiplying Power Series, we state the main result regarding multiplying power series, showing that if ∞∑n=0cnxn and ∞∑n=0dnxn converge on a common interval I, then we can multiply the series in this way, and the resulting series also converges on the interval I.
theorem: Multiplying Power Series
Suppose that the power series ∞∑n=0cnxn and ∞∑n=0dnxn converge to f and g, respectively, on a common interval I. Let
en=c0dn+c1dn−1+c2dn−2+⋯+cn−1d1+cnd0=n∑k=0ckdn−k.
Then
(∞∑n=0cnxn)(∞∑n=0dnxn)=∞∑n=0enxn
and
∞∑n=0enxnconverges tof(x)⋅g(x) on I.
The series ∞∑n=0enxn is known as the Cauchy product of the series ∞∑n=0cnxn and ∞∑n=0dnxn.
We omit the proof of this theorem, as it is beyond the level of this text and is typically covered in a more advanced course. We now provide an example of this theorem by finding the power series representation for
f(x)=1(1−x)(1−x2)
using the power series representations for
y=11−xandy=11−x2.
Example: Multiplying Power Series
Multiply the power series representation
11−x=∞∑n=0xn=1+x+x2+x3+⋯
for |x|<1 with the power series representation
11−x2=∞∑n=0(x2)n=1+x2+x4+x6+⋯
for |x|<1 to construct a power series for f(x)=1(1−x)(1−x2) on the interval (−1,1).
Show Solution
We need to multiply
(1+x+x2+x3+⋯)(1+x2+x4+x6+⋯).
Writing out the first several terms, we see that the product is given by
Since the series for y=11−x and y=11−x2 both converge on the interval (−1,1), the series for the product also converges on the interval (−1,1).
Watch the following video to see the worked solution to Example: Multiplying Power Series.
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