Worksheet: Representing Rational Functions Using Power Series

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)=𝑥3𝑥.

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

Q2:

Let us consider 𝑓(𝑥)=11+𝑥.

Find a power series representation for 𝑓(𝑥).

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

Find its interval of convergence.

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

Q3:

Consider (𝑥)=4𝑥1+𝑥.

Find a power series representation for (𝑥).

  • A ( 𝑥 ) = 4 ( 1 ) 𝑥
  • B ( 𝑥 ) = 4 ( 1 ) 𝑥
  • C ( 𝑥 ) = ( 1 ) 𝑥
  • D ( 𝑥 ) = 4 ( 𝑥 )
  • E ( 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)=12+𝑥.

Find the power series for 𝑓(𝑥).

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

Identify its interval of convergence.

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

Q6:

Consider the function 𝑓(𝑥)=119𝑥.

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 1 3 , 1 3
  • C 1 3 , 1 3
  • D 1 9 , 1 9
  • E 1 9 , 1 9

Q7:

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

Find the power series for 𝑓(𝑥).

  • A ( 1 ) 𝑥 3
  • B 𝑥 3
  • C ( 1 ) 𝑥 3
  • D 𝑥 3
  • E ( 1 ) 𝑥 3

Q8:

Consider the function 𝑓(𝑥)=1𝑥2𝑥+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).

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

Q10:

Consider the function 𝑓(𝑥)=𝑥19𝑥.

Find the power series for 𝑓(𝑥).

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

Identify its interval of convergence.

  • A ( 1 , 1 )
  • B 1 9 , 1 9
  • C 1 3 , 1 3
  • D 1 9 , 1 9
  • E 1 3 , 1 3

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