Lesson Worksheet: Representing Rational Functions Using Power Series Mathematics • 12th Grade

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(π‘₯)=π‘₯3βˆ’π‘₯.

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

  • A𝑔(π‘₯)=ο„šπ‘₯3∞
  • B𝑔(π‘₯)=ο„šο€»π‘₯3ο‡βˆžοŠοŠ²οŠ¦οŠ
  • C𝑔(π‘₯)=ο„šπ‘₯3∞
  • D𝑔(π‘₯)=ο„šο€»π‘₯3ο‡βˆžοŠοŠ²οŠ¦οŠοŠ°οŠ§
  • E𝑔(π‘₯)=ο„šπ‘₯3∞

Find its radius of convergence.

  • A|π‘₯|<3
  • B|π‘₯|>3
  • C|π‘₯|>1
  • D|π‘₯|<1
  • E|π‘₯|<13

Q2:

Let us consider 𝑓(π‘₯)=11+π‘₯.

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

  • A𝑓(π‘₯)=ο„š(βˆ’1)π‘₯∞
  • B𝑓(π‘₯)=ο„š(βˆ’1)π‘₯∞
  • C𝑓(π‘₯)=ο„šπ‘₯∞
  • D𝑓(π‘₯)=ο„š(βˆ’1)π‘₯∞
  • E𝑓(π‘₯)=ο„šπ‘₯∞

Find its interval of convergence.

  • A|π‘₯|<12
  • B|π‘₯|<1
  • C|π‘₯|>1
  • D|π‘₯|>0
  • E|π‘₯|<0

Q3:

Consider β„Ž(π‘₯)=4π‘₯1+π‘₯.

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

  • Aβ„Ž(π‘₯)=ο„š4(βˆ’1)π‘₯∞
  • Bβ„Ž(π‘₯)=ο„š4(βˆ’1)π‘₯∞
  • Cβ„Ž(π‘₯)=ο„š(βˆ’1)π‘₯∞
  • Dβ„Ž(π‘₯)=ο„š4(βˆ’π‘₯)∞
  • Eβ„Ž(π‘₯)=ο„š4(βˆ’1)π‘₯∞

Find its interval of convergence.

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

Q4:

Use a power series to represent π‘₯1+π‘₯.

  • AβˆžοŠοŠ²οŠ¦οŠ¨οŠοŠ°οŠ§ο„šπ‘₯
  • BβˆžοŠοŠ²οŠ¦οŠοŠ¨οŠοŠ°οŠ§ο„š(βˆ’1)π‘₯
  • CβˆžοŠοŠ²οŠ¦οŠ¨οŠο„šπ‘₯
  • DβˆžοŠοŠ²οŠ¦οŠ¨οŠοŠ°οŠ¨ο„šπ‘₯
  • EβˆžοŠοŠ²οŠ¦οŠοŠ¨οŠο„š(βˆ’1)π‘₯

Q5:

Consider the function 𝑓(π‘₯)=11βˆ’9π‘₯.

Find the power series for 𝑓(π‘₯).

  • AβˆžοŠοŠ²οŠ¦οŠ¨οŠοŠ¨οŠο„š(3)(π‘₯)
  • BβˆžοŠοŠ²οŠ¦οŠ¨οŠοŠ¨οŠο„š(9)(π‘₯)
  • CβˆžοŠοŠ²οŠ¦οŠ¨οŠοŠ¨οŠο„š(3)(βˆ’π‘₯)
  • DβˆžοŠοŠ²οŠ¦οŠοŠ¨οŠο„š(3)(π‘₯)
  • EβˆžοŠοŠ²οŠ¦οŠ¨οŠοŠ¨οŠο„š(βˆ’3)(π‘₯)

Identify its interval of convergence.

  • A(βˆ’1,1)
  • Bο”βˆ’13,13
  • Cο€Όβˆ’13,13
  • Dο€Όβˆ’19,19
  • Eο”βˆ’19,19

Q6:

Consider the function 𝑓(π‘₯)=12+π‘₯.

Find the power series for 𝑓(π‘₯).

  • AβˆžοŠοŠ²οŠ¦οŠοŠ°οŠ§οŠο„šο€Ό12(π‘₯)
  • BβˆžοŠοŠ²οŠ¦οŠοŠ°οŠ§οŠο„šο€Όβˆ’12(βˆ’π‘₯)
  • CβˆžοŠοŠ²οŠ¦οŠοŠο„šο€Ό12(βˆ’π‘₯)
  • DβˆžοŠοŠ²οŠ¦οŠοŠ°οŠ¨οŠο„šο€Ό12(π‘₯)
  • EβˆžοŠοŠ²οŠ¦οŠοŠ°οŠ§οŠο„šο€Ό12(βˆ’π‘₯)

Identify its interval of convergence.

  • A(βˆ’βˆž,∞)
  • B(βˆ’2,2)
  • C[βˆ’2,2]
  • D(0,1]
  • E(βˆ’1,1)

Q7:

Consider the function 𝑓(π‘₯)=π‘₯9+π‘₯ with the center π‘Ž=0.

Find the power series for 𝑓(π‘₯).

  • AβˆžοŠοŠ²οŠ¦οŠοŠ¨οŠοŠ°οŠ§ο„š(βˆ’1)ο€»π‘₯3
  • BβˆžοŠοŠ²οŠ¦οŠ¨οŠο„šο€»π‘₯3
  • CβˆžοŠοŠ²οŠ¦οŠοŠ¨οŠο„š(βˆ’1)ο€»π‘₯3
  • DβˆžοŠοŠ²οŠ¦οŠ¨οŠοŠ°οŠ¨ο„šο€»π‘₯3
  • EβˆžοŠοŠ²οŠ¦οŠοŠ¨οŠοŠ°οŠ¨ο„š(βˆ’1)ο€»π‘₯3

Q8:

Consider the function 𝑓(π‘₯)=1π‘₯βˆ’2π‘₯+2 with the center π‘Ž=1.

Find the power series for 𝑓(π‘₯).

  • AβˆžοŠοŠ²οŠ¦οŠοŠ¨οŠο„š(βˆ’1)(π‘₯)
  • BβˆžοŠοŠ²οŠ¦οŠ¨οŠο„š(π‘₯βˆ’1)
  • CβˆžοŠοŠ²οŠ¦οŠοŠο„š(βˆ’1)(π‘₯βˆ’1)
  • DβˆžοŠοŠ²οŠ¦οŠοŠ¨οŠο„š(βˆ’1)(π‘₯βˆ’1)
  • EβˆžοŠοŠ²οŠ¦οŠοŠ¨οŠοŠ°οŠ§ο„š(βˆ’1)(π‘₯βˆ’1)

Q9:

Consider the function 𝑓(π‘₯)=2π‘₯ with the center π‘Ž=1.

Find the power series for 𝑓(π‘₯).

  • A2ο„š(1βˆ’π‘₯)∞
  • BβˆžοŠοŠ²οŠ¦οŠο„š(1βˆ’π‘₯)
  • CβˆžοŠοŠ²οŠ¦οŠοŠο„š(βˆ’1)(1βˆ’π‘₯)
  • D2ο„š(βˆ’1)(1βˆ’π‘₯)∞
  • E2ο„š(1βˆ’π‘₯)∞

Q10:

Consider the function 𝑓(π‘₯)=π‘₯1βˆ’9π‘₯.

Find the power series for 𝑓(π‘₯).

  • AβˆžοŠοŠ²οŠ¦οŠ¨οŠοŠ¨οŠοŠ°οŠ§ο„š(βˆ’3)π‘₯
  • BβˆžοŠοŠ²οŠ¦οŠ¨οŠοŠ¨οŠοŠ°οŠ§ο„š3π‘₯
  • CβˆžοŠοŠ²οŠ¦οŠ¨οŠοŠ¨οŠοŠ°οŠ§ο„š9π‘₯
  • DβˆžοŠοŠ²οŠ¦οŠοŠ¨οŠο„š3π‘₯
  • EβˆžοŠοŠ²οŠ¦οŠ¨οŠοŠ¨οŠοŠ°οŠ§ο„š9(βˆ’π‘₯)

Identify its interval of convergence.

  • A(βˆ’1,1)
  • Bο”βˆ’19,19
  • Cο€Όβˆ’13,13
  • Dο€Όβˆ’19,19
  • Eο”βˆ’13,13

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