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).

  • 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 (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.

  • A𝑥
  • B(1)𝑥
  • C𝑥
  • D𝑥
  • E(1)𝑥

Q5:

Consider the function 𝑓(𝑥)=12+𝑥.

Find the power series for 𝑓(𝑥).

  • A12(𝑥)
  • B12(𝑥)
  • C12(𝑥)
  • D12(𝑥)
  • E12(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)
  • B13,13
  • C13,13
  • D19,19
  • E19,19

Q7:

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

Find the power series for 𝑓(𝑥).

  • A(1)𝑥3
  • B𝑥3
  • C(1)𝑥3
  • D𝑥3
  • E(1)𝑥3

Q8:

Consider the function 𝑓(𝑥)=1𝑥2𝑥+2 with the center 𝑎=1.

Find the power series for 𝑓(𝑥).

  • A(1)(𝑥)
  • B(𝑥1)
  • C(1)(𝑥1)
  • D(1)(𝑥1)
  • E(1)(𝑥1)

Q9:

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

Find the power series for 𝑓(𝑥).

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

Q10:

Consider the function 𝑓(𝑥)=𝑥19𝑥.

Find the power series for 𝑓(𝑥).

  • A(3)𝑥
  • B3𝑥
  • C9𝑥
  • D3𝑥
  • E9(𝑥)

Identify its interval of convergence.

  • A(1,1)
  • B19,19
  • C13,13
  • D19,19
  • E13,13

Q11:

Find a power series representation for 𝑓(𝑥)=𝑥𝑥+7.

  • A𝑓(𝑥)=(1)7𝑥
  • B𝑓(𝑥)=(1)7𝑥
  • C𝑓(𝑥)=17𝑥
  • D𝑓(𝑥)=17𝑥
  • E𝑓(𝑥)=(1)7𝑥

Q12:

Find a power series representation for 𝑥6𝑥.

  • A(1)𝑥6
  • B𝑥6
  • C𝑥6
  • D(1)𝑥6
  • E𝑥6

Q13:

Determine the interval of convergence for the power series of the function 𝑓(𝑥)=2𝑥2+6𝑥.

  • A13,13
  • B(3,3)
  • C[3,3]
  • D16,16
  • E13,13

Q14:

Find a power series representation for 𝑓(𝑥)=1𝑥4𝑥3, centered at 2.

  • A𝑓(𝑥)=(𝑥2)
  • B𝑓(𝑥)=𝑥4𝑥3
  • C𝑓(𝑥)=(𝑥+2)
  • D𝑓(𝑥)=(1)(𝑥+2)
  • E𝑓(𝑥)=(1)(𝑥+2)

Q15:

Write the first four terms of the power series of the function 𝑓(𝑥)=225𝑥.

  • A152𝑥+254𝑥1258𝑥
  • B52𝑥+254𝑥+1258𝑥
  • C1+52𝑥+52𝑥+52𝑥
  • D1+52𝑥+254𝑥+1258𝑥
  • E152𝑥+52𝑥52𝑥

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