Worksheet: Power Series Representation of a Rational Function

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In this worksheet, we will practice using a power series to represent a rational function and indicate the interval over which the series converges.

Q1:

Let us consider 𝑔 ( 𝑥 ) = 𝑥 3 − 𝑥 .

Find a power series representation for 𝑔 ( 𝑥 ) .

  • A 𝑔 ( 𝑥 ) = 𝑥 3 ∞
  • B 𝑔 ( 𝑥 ) = 𝑥 3 ∞
  • C 𝑔 ( 𝑥 ) = 𝑥 3 ∞
  • D 𝑔 ( 𝑥 ) = 𝑥 3 ∞
  • E 𝑔 ( 𝑥 ) = 𝑥 3 ∞

Find its radius of convergence.

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

Q2:

Let us consider 𝑓 ( 𝑥 ) = 1 1 + 𝑥 .

Find a power series representation for 𝑓 ( 𝑥 ) .

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

Find its interval of convergence.

  • A | 𝑥 | < 1
  • B | 𝑥 | > 0
  • C | 𝑥 | < 0
  • D | 𝑥 | > 1
  • E | 𝑥 | < 1 2

Q3:

Consider ℎ ( 𝑥 ) = 4 𝑥 1 + 𝑥 .

Find a power series representation for ℎ ( 𝑥 ) .

  • A ℎ ( 𝑥 ) = 4 ( − 1 ) 𝑥 ∞
  • B ℎ ( 𝑥 ) = ( − 1 ) 𝑥 ∞
  • C ℎ ( 𝑥 ) = 4 ( − 1 ) 𝑥 ∞
  • D ℎ ( 𝑥 ) = 4 ( − 1 ) 𝑥 ∞
  • E ℎ ( 𝑥 ) = 4 ( − 𝑥 ) ∞

Find its interval of convergence.

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

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