Lesson Worksheet: Maclaurin Series Mathematics • Higher Education

In this worksheet, we will practice finding Maclaurin series of a function and finding the radius of convergence of the series.

Q1:

Consider the function 𝑓(π‘₯)=𝑒.

Find 𝑓(π‘₯).

  • A𝑒π‘₯ln
  • Bπ‘’ο—οŠ±οŠ§
  • C𝑒
  • Dlnπ‘₯
  • E𝑒π‘₯ο—οŠ±οŠ§ln

Find 𝑓(π‘₯)(), where 𝑓() represents the 𝑛th derivative of 𝑓 with respect to π‘₯.

  • Aπ‘’ο—οŠ±οŠ
  • B𝑒
  • C𝑒π‘₯+𝑒(βˆ’1)(π‘›βˆ’2)!π‘₯ο—οŠ±οŠο—οŠ±οŠ§οŠ()ln for 𝑛>1
  • D𝑒π‘₯+𝑒(βˆ’1)(π‘›βˆ’2)!π‘₯ο—ο—οŠ()ln for 𝑛>1
  • E(βˆ’1)(π‘›βˆ’2)!π‘₯() for 𝑛>1

Hence, derive the Maclaurin series for 𝑒.

  • A𝑒=ο„šπ‘₯𝑛!ο—βˆžοŠοŠ²οŠ§οŠ
  • B𝑒=ο„šπ‘₯𝑛!ο—βˆžοŠοŠ²οŠ¦οŠ
  • C𝑒=ο„šπ‘’π‘›!ο—βˆžοŠοŠ²οŠ¦οŠ
  • D𝑒=ο„šπ‘“(π‘Ž)(π‘₯βˆ’π‘Ž)𝑛!ο—βˆžοŠοŠ²οŠ§()
  • E𝑒=ο„šπ‘“(π‘Ž)(π‘₯βˆ’π‘Ž)𝑛!ο—βˆžοŠοŠ²οŠ¦()

What is the radius of convergence 𝑅 of the Maclaurin series for 𝑒?

  • AIt does not converge.
  • B𝑅=𝑒
  • C𝑅=1
  • D𝑅=+∞
  • E𝑅=100

Q2:

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

What are the first four derivatives of 𝑓 with respect to π‘₯?

  • A𝑓′(π‘₯)=π‘₯cos, 𝑓′′(π‘₯)=π‘₯sin, 𝑓′′′(π‘₯)=βˆ’π‘₯cos, and 𝑓(π‘₯)=π‘₯(οŠͺ)sin
  • B𝑓′(π‘₯)=βˆ’π‘₯cos, 𝑓′′(π‘₯)=βˆ’π‘₯sin, 𝑓′′′(π‘₯)=π‘₯cos, and 𝑓(π‘₯)=π‘₯(οŠͺ)sin
  • C𝑓′(π‘₯)=π‘₯cos, 𝑓′′(π‘₯)=βˆ’π‘₯sin, 𝑓′′′(π‘₯)=βˆ’π‘₯cos, and 𝑓(π‘₯)=βˆ’π‘₯(οŠͺ)sin
  • D𝑓′(π‘₯)=βˆ’π‘₯cos, 𝑓′′(π‘₯)=π‘₯sin, 𝑓′′′(π‘₯)=βˆ’π‘₯cos, and 𝑓(π‘₯)=π‘₯(οŠͺ)sin
  • E𝑓′(π‘₯)=π‘₯cos, 𝑓′′(π‘₯)=βˆ’π‘₯sin, 𝑓′′′(π‘₯)=βˆ’π‘₯cos, and 𝑓(π‘₯)=π‘₯(οŠͺ)sin

Write the general form for the 𝑛th derivative of 𝑓 with respect to π‘₯.

  • A𝑓(π‘₯)=βˆ’ο€»π‘₯+π‘›πœ‹2()cos
  • B𝑓(π‘₯)=ο€»π‘₯+π‘›πœ‹2()sin
  • C𝑓(π‘₯)=βˆ’ο€»π‘₯+π‘›πœ‹2()sin
  • D𝑓(π‘₯)=(π‘₯+π‘›πœ‹)()sin
  • E𝑓(π‘₯)=ο€»π‘₯+π‘›πœ‹2()cos

Hence, derive the Maclaurin series for sinπ‘₯.

  • AβˆžοŠοŠ²οŠ¦οŠ¨οŠοŠ°οŠ§οŠ¨οŠοŠ°οŠ§ο„š(βˆ’1)π‘₯(2𝑛+1)!
  • BβˆžοŠοŠ²οŠ¦οŠοŠ¨οŠο„š(βˆ’1)π‘₯(2𝑛)!
  • CβˆžοŠοŠ²οŠ¦οŠοŠο„š(βˆ’1)π‘₯𝑛!
  • DβˆžοŠοŠ²οŠ¦οŠοŠ¨οŠοŠ°οŠ§ο„š(βˆ’1)π‘₯(2𝑛+1)!
  • EβˆžοŠοŠ²οŠ¦οŠ¨οŠοŠο„š(βˆ’1)π‘₯𝑛!

What is the radius 𝑅 of convergence of the Maclaurin series for sinπ‘₯?

  • A𝑅=+∞
  • B𝑅=πœ‹2
  • C𝑅=2πœ‹
  • D𝑅=1
  • E𝑅=πœ‹

Q3:

Find the Maclaurin series of cosh2π‘₯=𝑒+𝑒2οŠ¨ο—οŠ±οŠ¨ο—.

  • AβˆžοŠοŠ²οŠ¦οŠ¨οŠοŠ°οŠ§ο„š(2π‘₯)(2𝑛+1)!
  • BβˆžοŠοŠ²οŠ¦οŠ¨οŠο„š(2π‘₯)(2𝑛)!
  • CβˆžοŠοŠ²οŠ¦οŠ¨οŠο„š(2π‘₯)(2𝑛)
  • DβˆžοŠοŠ²οŠ¦οŠο„š(2π‘₯)𝑛!
  • EβˆžοŠοŠ²οŠ¦οŠ¨οŠοŠ°οŠ§ο„š(2π‘₯)(2𝑛+1)

Q4:

Consider the function 𝑓(π‘₯)=ο€Ή1+π‘₯ln.

Derive the Maclaurin series for 𝑓.

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

Using the Maclaurin series, find ln1.04 to 5 decimal places.

Q5:

If the Maclaurin series of the function 𝑓is 𝑓(π‘₯)=3βˆ’12π‘₯+56π‘₯βˆ’1126π‘₯+2180π‘₯+β‹―οŠ¨οŠ©οŠͺ, find 𝑓′′′(0).

  • A56
  • Bβˆ’3313
  • Cβˆ’1126
  • D6340
  • E5

Q6:

If the Maclaurin series of the function 𝑓 is 𝑓(π‘₯)=2βˆ’16π‘₯+524π‘₯βˆ’760π‘₯+380π‘₯+β‹―οŠ¨οŠ©οŠͺ, find the equation of the tangent to the curve of 𝑓 at π‘₯=0.

  • A𝑦=βˆ’16π‘₯+2
  • B𝑦=2π‘₯βˆ’16
  • C𝑦=βˆ’16π‘₯+524
  • D𝑦=16π‘₯+2
  • E𝑦=2π‘₯+524

Q7:

Find the Maclaurin series of π‘₯π‘’οŠ©οŠ¨ο—.

  • AβˆžοŠοŠ²οŠ¦οŠοŠο„š(βˆ’1)1𝑛!π‘₯
  • BβˆžοŠοŠ²οŠ¦οŠοŠοŠοŠ°οŠ©ο„š(βˆ’1)2𝑛!π‘₯
  • CβˆžοŠοŠ²οŠ¦οŠοŠοŠ°οŠ©ο„š2𝑛π‘₯
  • DβˆžοŠοŠ²οŠ¦οŠοŠοŠ°οŠ©ο„š2𝑛!π‘₯
  • EβˆžοŠοŠ²οŠ¦οŠοŠο„š2𝑛!π‘₯

Q8:

Find the Maclaurin series of arctan5π‘₯.

  • AβˆžοŠοŠ²οŠ¦οŠοŠ¨οŠοŠ°οŠ§ο„š(βˆ’1)(5π‘₯)(2𝑛+1)
  • BβˆžοŠοŠ²οŠ¦οŠ¨οŠοŠ°οŠ§ο„š(5π‘₯)(2𝑛+1)
  • CβˆžοŠοŠ²οŠ¦οŠοŠ¨οŠο„š(βˆ’1)(5π‘₯)(2𝑛)
  • DβˆžοŠοŠ²οŠ¦οŠοŠ¨οŠοŠ°οŠ§ο„š(βˆ’1)(5π‘₯)(2𝑛+1)!
  • EβˆžοŠοŠ²οŠ¦οŠ¨οŠο„š(5π‘₯)(2𝑛)!

Q9:

Find the Maclaurin series of 31+π‘₯. Write your answer in sigma notation.

  • AβˆžοŠοŠ²οŠ¦οŠ©οŠο„š(π‘₯)
  • B3ο„š(βˆ’1)(π‘₯)∞
  • CβˆžοŠοŠ²οŠ¦οŠοŠο„š(βˆ’1)(3π‘₯)
  • DβˆžοŠοŠ²οŠ¦οŠοŠ©οŠο„š(βˆ’1)(π‘₯)
  • E3ο„š(π‘₯)∞

Q10:

Find the radius of convergence for the Maclaurin series for 𝑓(π‘₯)=2π‘₯cos.

  • A12
  • B2πœ‹
  • C∞
  • D1
  • Eπœ‹

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