Combining and Multiplying Power Series

Learning Outcomes

  • Combine power series by addition or subtraction
  • 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)=11xf(x)=11x, we can find power series representations for related functions, such as

y=3x1x2 and y=1(x1)(x3)y=3x1x2 and y=1(x1)(x3).

 

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=0cnxnn=0cnxn and n=0dnxnn=0dnxn converge to the functions f and g, respectively, on a common interval I.

  1. The power series n=0(cnxn±dnxn)n=0(cnxn±dnxn) converges to f±gf±g on I.
  2. For any integer m0m0 and any real number b, the power series n=0bxmcnxnn=0bxmcnxn converges to bxmf(x)bxmf(x) on I.
  3. For any integer m0m0 and any real number b, the series n=0cn(bxm)nn=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=0cnxnn=0cnxn and n=0dnxnn=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=0cnxnn=0cnxn and n=0dnxnn=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

Nn=0(cnxn+dnxn)=Nn=0cnxn+Nn=0dnxn=SN(x)+TN(x).Nn=0(cnxn+dnxn)=Nn=0cnxn+Nn=0dnxn=SN(x)+TN(x).

 

Because

limN(SN(x)+TN(x))=limNSN(x)+limNTN(x)=f(x)+g(x),limN(SN(x)+TN(x))=limNSN(x)+limNTN(x)=f(x)+g(x),

 

we conclude that the series n=0(cnxn+dnxn)n=0(cnxn+dnxn) converges to f(x)+g(x)f(x)+g(x).

◼

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=0anxnn=0anxn is a power series whose interval of convergence is (1,1)(1,1), and suppose that n=0bnxnn=0bnxn is a power series whose interval of convergence is (2,2)(2,2).

  1. Find the interval of convergence of the series n=0(anxn+bnxn)n=0(anxn+bnxn).
  2. Find the interval of convergence of the series n=0an3nxnn=0an3nxn.

Watch the following video to see the worked solution to Example: Combining Power Series.

You can view the transcript for “6.2.1” here (opens in new window).

try it

Suppose that n=0anxnn=0anxn has an interval of convergence of (1,1)(1,1). Find the interval of convergence of n=0an(x2)n.

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)=11x and we use the fact that

11x=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)=11x 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.

  1. f(x)=3x1+x2
  2. f(x)=1(x1)(x3)

f(x)=3x(11(-x2)).



Using the power series representation for f(x)=11x 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 11x 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(1x)(x3) as the sum of two fractions. We have

 

1(x1)(x3)=-12x1+12x3=121x123x=121x161x3.



First, using part ii. of Combining Power Series, we obtain

121x=n=012xnfor|x|<1.



Then, using parts ii. and iii. of Combining Power Series, we have

161x3=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(x1)(x3)=n=0(12163n)xn



where the interval of convergence is (1,1).

try it

Use the series for f(x)=11x on |x|<1 to construct a series for 1(1x)(x2). Determine the interval of convergence.

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.

try it

Find the function represented by the power series n=013nxn. Determine its interval of convergence.

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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

n=0cnxn=c0+c1x+c2x2+

 

and

n=0dnxn=d0+d1x+d2x2+.

 

It appears that the product should satisfy

(n=0cnxn)(n=0dnxn)=(c0+c1x+c2x2+)(d0+d1x+d2x2+)=c0d0+(c1d0+c0d1)x+(c2d0+c1d1+c0d2)x2+.

 

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+c1dn1+c2dn2++cn1d1+cnd0=nk=0ckdnk.

 

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(1x)(1x2)

 

using the power series representations for

y=11xandy=11x2.

 

Example: Multiplying Power Series

Multiply the power series representation

11x=n=0xn=1+x+x2+x3+

 

for |x|<1 with the power series representation

11x2=n=0(x2)n=1+x2+x4+x6+

 

for |x|<1 to construct a power series for f(x)=1(1x)(1x2) on the interval (1,1).

Watch the following video to see the worked solution to Example: Multiplying Power Series.

For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.

You can view the transcript for this segmented clip of “6.2.3” here (opens in new window).

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Multiply the series 11x=n=0xn by itself to construct a series for 1(1x)(1x).

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