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Lesson: Power Series Representation of a Rational Function

In this lesson, we will learn how to use a power series to represent a rational function and indicate the interval over which the series converges.

Worksheet • 3 Questions

Q1:

Let us consider 𝑔 ( π‘₯ ) = π‘₯ 3 βˆ’ π‘₯ .

Find a power series representation for 𝑔 ( π‘₯ ) .

  • A 𝑔 ( π‘₯ ) = ο„š ο€» π‘₯ 3  ∞      
  • B 𝑔 ( π‘₯ ) = ο„š π‘₯ 3 ∞       
  • C 𝑔 ( π‘₯ ) = ο„š π‘₯ 3 ∞       
  • D 𝑔 ( π‘₯ ) = ο„š ο€» π‘₯ 3  ∞    
  • E 𝑔 ( π‘₯ ) = ο„š π‘₯ 3 ∞      

Find its radius of convergence.

  • A | π‘₯ | < 3
  • B | π‘₯ | < 1 3
  • C | π‘₯ | > 3
  • D | π‘₯ | > 1
  • E | π‘₯ | < 1

Q2:

Let us consider 𝑓 ( π‘₯ ) = 1 1 + π‘₯  .

Find a power series representation for 𝑓 ( π‘₯ ) .

  • A 𝑓 ( π‘₯ ) = ο„š ( βˆ’ 1 ) π‘₯ ∞      
  • B 𝑓 ( π‘₯ ) = ο„š π‘₯ ∞      
  • C 𝑓 ( π‘₯ ) = ο„š ( βˆ’ 1 ) π‘₯ ∞     
  • D 𝑓 ( π‘₯ ) = ο„š ( βˆ’ 1 ) π‘₯ ∞       
  • E 𝑓 ( π‘₯ ) = ο„š π‘₯ ∞     

Find its interval of convergence.

  • A | π‘₯ | < 1
  • B | π‘₯ | < 1 2
  • C | π‘₯ | > 1
  • D | π‘₯ | > 0
  • E | π‘₯ | < 0

Q3:

Consider β„Ž ( π‘₯ ) = 4 π‘₯ 1 + π‘₯   .

Find a power series representation for β„Ž ( π‘₯ ) .

  • A β„Ž ( π‘₯ ) = ο„š 4 ( βˆ’ 1 ) π‘₯ ∞        
  • B β„Ž ( π‘₯ ) = ο„š 4 ( βˆ’ π‘₯ ) ∞       
  • C β„Ž ( π‘₯ ) = ο„š ( βˆ’ 1 ) π‘₯ ∞        
  • D β„Ž ( π‘₯ ) = ο„š 4 ( βˆ’ 1 ) π‘₯ ∞       
  • E β„Ž ( π‘₯ ) = ο„š 4 ( βˆ’ 1 ) π‘₯ ∞      

Find its interval of convergence.

  • A | π‘₯ | < 1
  • B | π‘₯ | < 4
  • C | π‘₯ | > 1
  • D | π‘₯ | > 0
  • E | π‘₯ | < 0
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