Worksheet: Maclaurin Series

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(𝑥)=𝑒.

Find 𝑓(𝑥).

  • A𝑒𝑥ln
  • B𝑒
  • C𝑒
  • Dln𝑥
  • E𝑒𝑥ln

Find 𝑓(𝑥)(), where 𝑓() represents the 𝑛th derivative of 𝑓 with respect to 𝑥.

  • A𝑒
  • B𝑒
  • C𝑒𝑥+𝑒(−1)(𝑛−2)!𝑥()ln for 𝑛>1
  • D𝑒𝑥+𝑒(−1)(𝑛−2)!𝑥()ln for 𝑛>1
  • E(−1)(𝑛−2)!𝑥() for 𝑛>1

Hence, derive the Maclaurin series for 𝑒.

  • A𝑒=𝑥𝑛!∞
  • B𝑒=𝑥𝑛!∞
  • C𝑒=𝑒𝑛!∞
  • D𝑒=𝑓(𝑎)(𝑥−𝑎)𝑛!∞()
  • E𝑒=𝑓(𝑎)(𝑥−𝑎)𝑛!∞()

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(𝑥)=𝑥sin.

What are the first four derivatives of 𝑓 with respect to 𝑥?

  • A𝑓′(𝑥)=𝑥cos, 𝑓′′(𝑥)=𝑥sin, 𝑓′′′(𝑥)=−𝑥cos, and 𝑓(𝑥)=𝑥()sin
  • B𝑓′(𝑥)=−𝑥cos, 𝑓′′(𝑥)=−𝑥sin, 𝑓′′′(𝑥)=𝑥cos, and 𝑓(𝑥)=𝑥()sin
  • C𝑓′(𝑥)=𝑥cos, 𝑓′′(𝑥)=−𝑥sin, 𝑓′′′(𝑥)=−𝑥cos, and 𝑓(𝑥)=−𝑥()sin
  • D𝑓′(𝑥)=−𝑥cos, 𝑓′′(𝑥)=𝑥sin, 𝑓′′′(𝑥)=−𝑥cos, and 𝑓(𝑥)=𝑥()sin
  • E𝑓′(𝑥)=𝑥cos, 𝑓′′(𝑥)=−𝑥sin, 𝑓′′′(𝑥)=−𝑥cos, and 𝑓(𝑥)=𝑥()sin

Write the general form for the 𝑛th derivative of 𝑓 with respect to 𝑥.

  • A𝑓(𝑥)=−𝑥+𝑛𝜋2()cos
  • B𝑓(𝑥)=𝑥+𝑛𝜋2()sin
  • C𝑓(𝑥)=−𝑥+𝑛𝜋2()sin
  • D𝑓(𝑥)=(𝑥+𝑛𝜋)()sin
  • E𝑓(𝑥)=𝑥+𝑛𝜋2()cos

Hence, derive the Maclaurin series for sin𝑥.

  • A∞(−1)𝑥(2𝑛+1)!
  • B∞(−1)𝑥(2𝑛)!
  • C∞(−1)𝑥𝑛!
  • D∞(−1)𝑥(2𝑛+1)!
  • E∞(−1)𝑥𝑛!

What is the radius 𝑅 of convergence of the Maclaurin series for sin𝑥?

  • A𝑅=+∞
  • B𝑅=𝜋2
  • C𝑅=2𝜋
  • D𝑅=1
  • E𝑅=𝜋

Q3:

Consider the function 𝑓(𝑥)=𝑥cos.

What are the first four derivatives of 𝑓 with respect to 𝑥?

  • A𝑓′(𝑥)=−𝑥sin, 𝑓′′(𝑥)=𝑥cos, 𝑓′′′(𝑥)=−𝑥sin, and 𝑓(𝑥)=𝑥()cos
  • B𝑓′(𝑥)=𝑥cos, 𝑓′′(𝑥)=−𝑥sin, 𝑓′′′(𝑥)=−𝑥cos, and 𝑓(𝑥)=𝑥()sin
  • C𝑓′(𝑥)=−𝑥sin, 𝑓′′(𝑥)=−𝑥cos, 𝑓′′′(𝑥)=𝑥sin, and 𝑓(𝑥)=𝑥()cos
  • D𝑓′(𝑥)=𝑥sin, 𝑓′′(𝑥)=𝑥cos, 𝑓′′′(𝑥)=−𝑥sin, and 𝑓(𝑥)=−𝑥()cos
  • E𝑓′(𝑥)=𝑥sin, 𝑓′′(𝑥)=𝑥cos, 𝑓′′′(𝑥)=𝑥sin, and 𝑓(𝑥)=𝑥()cos

Write the general form for the 𝑛th derivative of 𝑓 with respect to 𝑥.

  • A𝑓(𝑥)=𝑥+𝑛𝜋2()sin
  • B𝑓(𝑥)=−𝑥+𝑛𝜋2()cos
  • C𝑓(𝑥)=𝑥+𝑛𝜋2()cos
  • D𝑓(𝑥)=(𝑥+𝑛𝜋)()cos
  • E𝑓(𝑥)=−𝑥+𝑛𝜋2()sin

Hence, derive the Maclaurin series for cos𝑥.

  • A∞(−1)𝑥(2𝑛)!
  • B∞(−1)𝑥𝑛!
  • C∞(−1)𝑥(2𝑛+1)!
  • D∞(−1)𝑥𝑛!
  • E∞(−1)𝑥(2𝑛)!

What is the radius 𝑅 of convergence of the Maclaurin series for cos𝑥?

  • A𝑅=+∞
  • B𝑅=𝜋2
  • C𝑅=2𝜋
  • D𝑅=1
  • E𝑅=𝜋

Q4:

Find the Maclaurin series of cosh2𝑥=𝑒+𝑒2.

  • A∞(2𝑥)(2𝑛+1)!
  • B∞(2𝑥)(2𝑛)!
  • C∞(2𝑥)(2𝑛)
  • D∞(2𝑥)𝑛!
  • E∞(2𝑥)(2𝑛+1)

Q5:

Consider the function 𝑓(𝑥)=1+𝑥ln.

Derive the Maclaurin series for 𝑓.

  • A∞(−1)𝑥𝑛
  • B∞(−1)𝑥𝑛
  • C∞𝑥𝑛!
  • D∞(−1)𝑥𝑛!
  • E∞𝑥𝑛!

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

Q6:

Find the Maclaurin series of 31+𝑥. Write your answer in sigma notation.

  • A∞(𝑥)
  • B3(−1)(𝑥)∞
  • C∞(−1)(3𝑥)
  • D∞(−1)(𝑥)
  • E3(𝑥)∞

Q7:

Find the Maclaurin series of arctan5𝑥.

  • A∞(−1)(5𝑥)(2𝑛+1)
  • B∞(5𝑥)(2𝑛+1)
  • C∞(−1)(5𝑥)(2𝑛)
  • D∞(−1)(5𝑥)(2𝑛+1)!
  • E∞(5𝑥)(2𝑛)!

Q8:

Find the Maclaurin series of 𝑥𝑒.

  • A∞(−1)1𝑛!𝑥
  • B∞(−1)2𝑛!𝑥
  • C∞2𝑛𝑥
  • D∞2𝑛!𝑥
  • E∞2𝑛!𝑥

Q9:

If the Maclaurin series of the function 𝑓is 𝑓(𝑥)=3−12𝑥+56𝑥−1126𝑥+2180𝑥+⋯, find 𝑓′′′(0).

  • A56
  • B−3313
  • C−1126
  • D6340
  • E5

Q10:

If the Maclaurin series of the function 𝑓 is 𝑓(𝑥)=2−16𝑥+524𝑥−760𝑥+380𝑥+⋯, find the equation of the tangent to the curve of 𝑓 at 𝑥=0.

  • A𝑦=−16𝑥+2
  • B𝑦=2𝑥−16
  • C𝑦=−16𝑥+524
  • D𝑦=16𝑥+2
  • E𝑦=2𝑥+524

Q11:

Find the radius of convergence for the Maclaurin series for 𝑓(𝑥)=2𝑥cos.

  • A12
  • B2𝜋
  • C∞
  • D1
  • E𝜋

Q12:

Write the first four nonzero terms of the Maclaurin expansion for 𝑓(𝑥)=11𝑥𝑒 in ascending powers of 𝑥.

  • A22𝑥+44𝑥+44𝑥+88𝑥3
  • B11𝑥+22𝑥+22𝑥+44𝑥3
  • C−22𝑥−44𝑥−44𝑥−88𝑥3
  • D11𝑥+22𝑥+22𝑥+44𝑥3
  • E−11𝑥−22𝑥−22𝑥−44𝑥3

Q13:

Find the Maclaurin series for 𝑓(𝑥)=110𝑥+1.

  • A∞(10)𝑥
  • B∞(−10)𝑥
  • C∞(10)𝑥
  • D∞(−10)𝑥
  • E∞(−10)𝑥

Q14:

Find the radius of convergence for the Maclaurin series for 𝑓(𝑥)=110𝑥+1.

  • A14
  • B12
  • C110
  • D10
  • E1

Q15:

By calculating the Maclaurin series for 𝑓(𝑥)=2𝑥sin and 𝑔(𝑥)=𝑒, or otherwise, find the Maclaurin series for 𝑓(𝑥)+𝑔(𝑥).

  • A∞2𝑥1𝑘!−(−2)𝑥(1+𝑘)!
  • B∞2𝑥1𝑘!−(−2)𝑥(1+2𝑘)!
  • C∞(−2)𝑥1𝑘!−(2)𝑥(1+2𝑘)!
  • D∞(−2)𝑥1𝑘!−(−2)𝑥(1+2𝑘)!
  • E∞2𝑥1𝑘!−(2)𝑥(1+2𝑘)!

Q16:

A Maclaurin series is given by ∞(−1)(𝑛)𝑥. Find the radius of convergence for the series.

  • A13
  • B12
  • C2
  • D1
  • E+∞

Q17:

Find the Maclaurin series of the function 𝑓(𝑥)=𝑒.

  • A𝑓(𝑥)=1+𝑥+12𝑥+16𝑥+⋯
  • B𝑓(𝑥)=1+𝑥+12𝑥+13𝑥+⋯
  • C𝑓(𝑥)=1+𝑥+12𝑥+14𝑥+⋯
  • D𝑓(𝑥)=1−𝑥+12𝑥−16𝑥+⋯
  • E𝑓(𝑥)=1−𝑥+12𝑥−13𝑥+⋯

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