Worksheet: Operations on Power Series

In this worksheet, we will practice adding, subtracting, and multiplying two power series and finding the radius of convergence of the resulting power series.

Q1:

Suppose that 𝑎𝑥 is a power series whose interval of convergence is (3,3) and that 𝑏𝑥 is a power series whose interval of convergence is (5,5).

Find the interval of convergence of the series (𝑎𝑥𝑏𝑥).

  • A(3,5)
  • B(5,5)
  • C(8,8)
  • D(3,3)
  • E(5,3)

Find the interval of convergence of the series 𝑏2𝑥.

  • A(6,6)
  • B(5,5)
  • C52,52
  • D(3,3)
  • E(10,10)

Q2:

Use partial fractions to find the power series of the function 𝑓(𝑥)=3(𝑥2)(𝑥+1).

  • A(1)12𝑥
  • B(2)(1)𝑥
  • C(1)+(2)𝑥
  • D(1)+12𝑥
  • E(1)12𝑥

Q3:

Consider the power series 3𝑥.

Find the function 𝑓 represented by this series.

  • A𝑓(𝑥)=113𝑥
  • B𝑓(𝑥)=13+𝑥
  • C𝑓(𝑥)=13𝑥
  • D𝑓(𝑥)=11+3𝑥
  • E𝑓(𝑥)=13𝑥1

Determine the interval of convergence of the series.

  • A13,13
  • B1,13
  • C(3,3)
  • D(1,1)
  • E12,12

Q4:

Consider the functions 𝑓(𝑥)=(𝑥1)𝑛! and 𝑔(𝑥)=(1)(𝑥1)𝑛!.

Find the power series of 12[𝑓(𝑥)+𝑔(𝑥)].

  • A(1)(𝑥1)2𝑛!
  • B(𝑥1)2𝑛!
  • C(𝑥1)(2𝑛+1)!
  • D(𝑥1)(2𝑛+1)!
  • E(1)(𝑥1)2𝑛!

Find the power series of 12[𝑓(𝑥)𝑔(𝑥)].

  • A(1)(𝑥1)2𝑛!
  • B(1)(𝑥1)2𝑛!
  • C(𝑥1)(2𝑛+1)!
  • D(𝑥1)(2𝑛+1)!
  • E(𝑥1)2𝑛!

Q5:

Let 𝑓(𝑥)=1(𝑥1)(𝑥2).

Construct a power series for the function 𝑓(𝑥).

  • A112𝑥
  • B1+12𝑥
  • C12+1𝑥
  • D112𝑥
  • E121𝑥

Find the interval of convergence of the power series.

  • A(1,1)
  • B32,1
  • C(2,2)
  • D1,32
  • E32,32

Q6:

Use partial fractions to find the power series of the function 𝑓(𝑥)=3(𝑥+1)(𝑥+4).

  • A114𝑥
  • B(1)114𝑥
  • C(1)114𝑥
  • D114𝑥
  • E114𝑥

Q7:

Consider the functions 𝑓(𝑥)=(2𝑥)(2𝑛)! and 𝑔(𝑥)=(2𝑥)(2𝑛+1)!.

Find the power series of 𝑓(𝑥)+𝑔(𝑥).

  • A(2𝑥)(𝑛)
  • B(1)(2𝑥)(𝑛+1)
  • C𝑥(𝑛)!
  • D(2𝑥)(𝑛)!
  • E(1)(2𝑥)(𝑛)!

Find the power series of 𝑓(𝑥)𝑔(𝑥).

  • A(1)(2𝑥)(𝑛+1)
  • B(2𝑥)(𝑛)
  • C𝑥(𝑛)!
  • D(1)(2𝑥)(𝑛)!
  • E(2𝑥)(𝑛)!

Q8:

Multiply the series 11+𝑥=(1)𝑥 by itself to construct a series for 1(1+𝑥). Write the answer in sigma notation.

  • A(𝑛+1)𝑥
  • B(1)(𝑛+1)𝑥
  • C𝑥
  • D𝑛(1)𝑥
  • E(1)𝑥

Q9:

Consider the power series 𝑓(𝑥)=14𝑥.

Find the function 𝑓 represented by this series.

  • A𝑓(𝑥)=4𝑥4
  • B𝑓(𝑥)=44+𝑥
  • C𝑓(𝑥)=44𝑥
  • D𝑓(𝑥)=144𝑥
  • E𝑓(𝑥)=1𝑥+4

Determine the interval of convergence of the series.

  • A(1,1)
  • B14,1
  • C(4,4)
  • D14,14
  • E4,14

Let 𝑔(𝑥)=𝑓(𝑥). Find 𝑓(𝑥)+𝑔(𝑥) in sigma notation.

  • A8(1)14𝑥
  • B(1)14𝑥
  • C12𝑥
  • D214𝑥
  • E814𝑥

Q10:

Find 𝑥(2𝑥).

  • A21𝑥
  • B(2+1)𝑥
  • C21𝑥
  • D(2)𝑥
  • E2+1𝑥

Q11:

Consider the power series representations of 𝑓(𝑥)=11𝑥 and 𝑔(𝑥)=11+𝑥. Use them, or otherwise, to calculate the first three nonzero terms, in ascending powers of 𝑥, for the power series of 𝑓(𝑥)+𝑔(𝑥).

  • A2𝑥+2𝑥+2𝑥
  • B2𝑥+2𝑥+2𝑥
  • C2+2𝑥+2𝑥
  • D2+2𝑥+2𝑥
  • E22𝑥2𝑥

Q12:

Consider the power series 𝑓(𝑥)=(𝑥)𝑛. If 𝑔(𝑥)=𝑓(𝑥), what is the radius of convergence of 𝑓(𝑥)+𝑔(𝑥)?

  • A1
  • B12
  • C0
  • D
  • E2

Q13:

Using partial fractions, calculate the power series of 𝑓(𝑥)=𝑥𝑥1.

  • A12(1)1𝑥
  • B2𝑥
  • C(1)1𝑥
  • D14(1)1𝑥
  • E122𝑥

Q14:

Consider the power series representations of 𝑓(𝑥)=𝑒 and 𝑔(𝑥)=11𝑥. Use them to calculate the first four nonzero terms, in ascending powers of 𝑥, for the power series of 𝑒1𝑥.

  • A12𝑥+5𝑥28𝑥3
  • B1+2𝑥+5𝑥2+8𝑥3
  • C1+2𝑥+𝑥2+𝑥3
  • D1+2𝑥5𝑥2+8𝑥3
  • E12𝑥5𝑥28𝑥3

Q15:

Consider the power series 𝑓(𝑥)=𝑥𝑛 and 𝑔(𝑥)=𝑛𝑥. Find the first three nonzero terms, in ascending powers of 𝑥, of the power series that represents 𝑓(𝑥)𝑔(𝑥).

  • A1+18𝑥+8227
  • B1+2𝑥+134
  • C1+2𝑥+8227
  • D1+18𝑥+134
  • E1+178𝑥+355108𝑥

Q16:

Consider the power series of 𝑓(𝑥)=𝑥sin. By calculating 𝑓(𝑥)𝑓(𝑥), calculate the first three nonzero terms, in ascending powers of 𝑥, for the power series of 𝑔(𝑥)=𝑥sin.

  • A𝑥+𝑥32𝑥45
  • B𝑥+𝑥3+2𝑥45
  • C𝑥+𝑥32𝑥45
  • D𝑥𝑥3+2𝑥45
  • E𝑥𝑥3+2𝑥45

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