Worksheet: Maclaurin Series

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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).

  • A 𝑒 𝑥 l n
  • B 𝑒
  • C 𝑒
  • D l n 𝑥
  • E 𝑒 𝑥 l n

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

  • A 𝑒
  • B 𝑒
  • C 𝑒 𝑥 + 𝑒 ( − 1 ) ( 𝑛 − 2 ) ! 𝑥 ( ) l n for 𝑛>1
  • D 𝑒 𝑥 + 𝑒 ( − 1 ) ( 𝑛 − 2 ) ! 𝑥 ( ) l n for 𝑛>1
  • E ( − 1 ) ( 𝑛 − 2 ) ! 𝑥 ( ) for 𝑛>1

Hence, derive the Maclaurin series for 𝑒.

  • A 𝑒 = 𝑥 𝑛 ! ∞
  • B 𝑒 = 𝑥 𝑛 ! ∞
  • C 𝑒 = 𝑒 𝑛 ! ∞
  • D 𝑒 = 𝑓 ( 𝑎 ) ( 𝑥 − 𝑎 ) 𝑛 ! ∞ ( )
  • E 𝑒 = 𝑓 ( 𝑎 ) ( 𝑥 − 𝑎 ) 𝑛 ! ∞ ( )

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

  • AIt does not converge.
  • B 𝑅 = 𝑒
  • C 𝑅 = 1
  • D 𝑅 = + ∞
  • E 𝑅 = 1 0 0

Q2:

Consider the function 𝑓(𝑥)=𝑥sin.

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

  • A 𝑓 ′ ( 𝑥 ) = 𝑥 c o s , 𝑓 ′ ′ ( 𝑥 ) = 𝑥 s i n , 𝑓 ′ ′ ′ ( 𝑥 ) = − 𝑥 c o s , and 𝑓(𝑥)=𝑥()sin
  • B 𝑓 ′ ( 𝑥 ) = − 𝑥 c o s , 𝑓 ′ ′ ( 𝑥 ) = − 𝑥 s i n , 𝑓 ′ ′ ′ ( 𝑥 ) = 𝑥 c o s , and 𝑓(𝑥)=𝑥()sin
  • C 𝑓 ′ ( 𝑥 ) = 𝑥 c o s , 𝑓 ′ ′ ( 𝑥 ) = − 𝑥 s i n , 𝑓 ′ ′ ′ ( 𝑥 ) = − 𝑥 c o s , and 𝑓(𝑥)=−𝑥()sin
  • D 𝑓 ′ ( 𝑥 ) = − 𝑥 c o s , 𝑓 ′ ′ ( 𝑥 ) = 𝑥 s i n , 𝑓 ′ ′ ′ ( 𝑥 ) = − 𝑥 c o s , and 𝑓(𝑥)=𝑥()sin
  • E 𝑓 ′ ( 𝑥 ) = 𝑥 c o s , 𝑓 ′ ′ ( 𝑥 ) = − 𝑥 s i n , 𝑓 ′ ′ ′ ( 𝑥 ) = − 𝑥 c o s , and 𝑓(𝑥)=𝑥()sin

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

  • A 𝑓 ( 𝑥 ) = − 𝑥 + 𝑛 𝜋 2 ( ) c o s
  • B 𝑓 ( 𝑥 ) = 𝑥 + 𝑛 𝜋 2 ( ) s i n
  • C 𝑓 ( 𝑥 ) = − 𝑥 + 𝑛 𝜋 2 ( ) s i n
  • D 𝑓 ( 𝑥 ) = ( 𝑥 + 𝑛 𝜋 ) ( ) s i n
  • E 𝑓 ( 𝑥 ) = 𝑥 + 𝑛 𝜋 2 ( ) c o s

Hence, derive the Maclaurin series for sin𝑥.

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

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

  • A 𝑅 = + ∞
  • B 𝑅 = 𝜋 2
  • C 𝑅 = 2 𝜋
  • D 𝑅 = 1
  • E 𝑅 = 𝜋

Q3:

Consider the function 𝑓(𝑥)=𝑥cos.

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

  • A 𝑓 ′ ( 𝑥 ) = − 𝑥 s i n , 𝑓 ′ ′ ( 𝑥 ) = 𝑥 c o s , 𝑓 ′ ′ ′ ( 𝑥 ) = − 𝑥 s i n , and 𝑓(𝑥)=𝑥()cos
  • B 𝑓 ′ ( 𝑥 ) = 𝑥 c o s , 𝑓 ′ ′ ( 𝑥 ) = − 𝑥 s i n , 𝑓 ′ ′ ′ ( 𝑥 ) = − 𝑥 c o s , and 𝑓(𝑥)=𝑥()sin
  • C 𝑓 ′ ( 𝑥 ) = − 𝑥 s i n , 𝑓 ′ ′ ( 𝑥 ) = − 𝑥 c o s , 𝑓 ′ ′ ′ ( 𝑥 ) = 𝑥 s i n , and 𝑓(𝑥)=𝑥()cos
  • D 𝑓 ′ ( 𝑥 ) = 𝑥 s i n , 𝑓 ′ ′ ( 𝑥 ) = 𝑥 c o s , 𝑓 ′ ′ ′ ( 𝑥 ) = − 𝑥 s i n , and 𝑓(𝑥)=−𝑥()cos
  • E 𝑓 ′ ( 𝑥 ) = 𝑥 s i n , 𝑓 ′ ′ ( 𝑥 ) = 𝑥 c o s , 𝑓 ′ ′ ′ ( 𝑥 ) = 𝑥 s i n , and 𝑓(𝑥)=𝑥()cos

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

  • A 𝑓 ( 𝑥 ) = 𝑥 + 𝑛 𝜋 2 ( ) s i n
  • B 𝑓 ( 𝑥 ) = − 𝑥 + 𝑛 𝜋 2 ( ) c o s
  • C 𝑓 ( 𝑥 ) = 𝑥 + 𝑛 𝜋 2 ( ) c o s
  • D 𝑓 ( 𝑥 ) = ( 𝑥 + 𝑛 𝜋 ) ( ) c o s
  • E 𝑓 ( 𝑥 ) = − 𝑥 + 𝑛 𝜋 2 ( ) s i n

Hence, derive the Maclaurin series for cos𝑥.

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

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

  • A 𝑅 = + ∞
  • B 𝑅 = 𝜋 2
  • C 𝑅 = 2 𝜋
  • D 𝑅 = 1
  • E 𝑅 = 𝜋

Q4:

Find the Maclaurin series of cosh2𝑥=𝑒+𝑒2.

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

Q5:

Consider the function 𝑓(𝑥)=1+𝑥ln.

Derive the Maclaurin series for 𝑓.

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

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 )
  • B 3 ( − 1 ) ( 𝑥 ) ∞
  • C ∞ ( − 1 ) ( 3 𝑥 )
  • D ∞ ( − 1 ) ( 𝑥 )
  • E 3 ( 𝑥 ) ∞

Q7:

Find the Maclaurin series of arctan5𝑥.

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

Q8:

Find the Maclaurin series of 𝑥𝑒.

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

Q9:

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

  • A 5 6
  • B − 3 3 1 3
  • C − 1 1 2 6
  • D 6 3 4 0
  • 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.

  • A 𝑦 = − 1 6 𝑥 + 2
  • B 𝑦 = 2 𝑥 − 1 6
  • C 𝑦 = − 1 6 𝑥 + 5 2 4
  • D 𝑦 = 1 6 𝑥 + 2
  • E 𝑦 = 2 𝑥 + 5 2 4

Q11:

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

  • A 1 2
  • B 2 𝜋
  • C ∞
  • D1
  • E 𝜋

Q12:

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

  • A 2 2 𝑥 + 4 4 𝑥 + 4 4 𝑥 + 8 8 𝑥 3
  • B 1 1 𝑥 + 2 2 𝑥 + 2 2 𝑥 + 4 4 𝑥 3
  • C − 2 2 𝑥 − 4 4 𝑥 − 4 4 𝑥 − 8 8 𝑥 3
  • D 1 1 𝑥 + 2 2 𝑥 + 2 2 𝑥 + 4 4 𝑥 3
  • E − 1 1 𝑥 − 2 2 𝑥 − 2 2 𝑥 − 4 4 𝑥 3

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