Worksheet: Power Series Representation of a Rational Function

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 ( 𝑥 ) = 4 𝑥 1 + 𝑥 .

Find a power series representation for ( 𝑥 ) .

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

Find its interval of convergence.

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

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.