Lesson Worksheet: Representing Rational Functions Using Power Series Mathematics • 12th Grade

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In this worksheet, we will practice using the formula of the infinite sum of geometric series to find a power series representation for some rational functions.

Q1:

Let us consider 𝑔(𝑥)=𝑥3−𝑥.

Find a power series representation for 𝑔(𝑥).

  • A𝑔(𝑥)=𝑥3∞
  • B𝑔(𝑥)=𝑥3∞
  • C𝑔(𝑥)=𝑥3∞
  • D𝑔(𝑥)=𝑥3∞
  • E𝑔(𝑥)=𝑥3∞

Find its radius of convergence.

  • A|𝑥|<3
  • B|𝑥|>3
  • C|𝑥|>1
  • D|𝑥|<1
  • E|𝑥|<13

Q2:

Let us consider 𝑓(𝑥)=11+𝑥.

Find a power series representation for 𝑓(𝑥).

  • A𝑓(𝑥)=(−1)𝑥∞
  • B𝑓(𝑥)=(−1)𝑥∞
  • C𝑓(𝑥)=𝑥∞
  • D𝑓(𝑥)=(−1)𝑥∞
  • E𝑓(𝑥)=𝑥∞

Find its interval of convergence.

  • A|𝑥|<12
  • B|𝑥|<1
  • C|𝑥|>1
  • D|𝑥|>0
  • E|𝑥|<0

Q3:

Consider ℎ(𝑥)=4𝑥1+𝑥.

Find a power series representation for ℎ(𝑥).

  • Aℎ(𝑥)=4(−1)𝑥∞
  • Bℎ(𝑥)=4(−1)𝑥∞
  • Cℎ(𝑥)=(−1)𝑥∞
  • Dℎ(𝑥)=4(−𝑥)∞
  • Eℎ(𝑥)=4(−1)𝑥∞

Find its interval of convergence.

  • A|𝑥|>1
  • B|𝑥|<4
  • C|𝑥|>0
  • D|𝑥|<1
  • E|𝑥|<0

Q4:

Use a power series to represent 𝑥1+𝑥.

  • A∞𝑥
  • B∞(−1)𝑥
  • C∞𝑥
  • D∞𝑥
  • E∞(−1)𝑥

Q5:

Consider the function 𝑓(𝑥)=11−9𝑥.

Find the power series for 𝑓(𝑥).

  • A∞(3)(𝑥)
  • B∞(9)(𝑥)
  • C∞(3)(−𝑥)
  • D∞(3)(𝑥)
  • E∞(−3)(𝑥)

Identify its interval of convergence.

  • A(−1,1)
  • B−13,13
  • C−13,13
  • D−19,19
  • E−19,19

Q6:

Consider the function 𝑓(𝑥)=12+𝑥.

Find the power series for 𝑓(𝑥).

  • A∞12(𝑥)
  • B∞−12(−𝑥)
  • C∞12(−𝑥)
  • D∞12(𝑥)
  • E∞12(−𝑥)

Identify its interval of convergence.

  • A(−∞,∞)
  • B(−2,2)
  • C[−2,2]
  • D(0,1]
  • E(−1,1)

Q7:

Consider the function 𝑓(𝑥)=𝑥9+𝑥 with the center 𝑎=0.

Find the power series for 𝑓(𝑥).

  • A∞(−1)𝑥3
  • B∞𝑥3
  • C∞(−1)𝑥3
  • D∞𝑥3
  • E∞(−1)𝑥3

Q8:

Consider the function 𝑓(𝑥)=1𝑥−2𝑥+2 with the center 𝑎=1.

Find the power series for 𝑓(𝑥).

  • A∞(−1)(𝑥)
  • B∞(𝑥−1)
  • C∞(−1)(𝑥−1)
  • D∞(−1)(𝑥−1)
  • E∞(−1)(𝑥−1)

Q9:

Consider the function 𝑓(𝑥)=2𝑥 with the center 𝑎=1.

Find the power series for 𝑓(𝑥).

  • A2(1−𝑥)∞
  • B∞(1−𝑥)
  • C∞(−1)(1−𝑥)
  • D2(−1)(1−𝑥)∞
  • E2(1−𝑥)∞

Q10:

Consider the function 𝑓(𝑥)=𝑥1−9𝑥.

Find the power series for 𝑓(𝑥).

  • A∞(−3)𝑥
  • B∞3𝑥
  • C∞9𝑥
  • D∞3𝑥
  • E∞9(−𝑥)

Identify its interval of convergence.

  • A(−1,1)
  • B−19,19
  • C−13,13
  • D−19,19
  • E−13,13

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