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 𝑥 l n
  • D 𝑒
  • E l n 𝑥

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

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

Hence, derive the Maclaurin series for 𝑒 .

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

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

  • A 𝑅 = + ∞
  • B 𝑅 = 1 0 0
  • C 𝑅 = 1
  • D 𝑅 = 𝑒
  • EIt does not converge.

Q2:

Consider the function 𝑓 ( 𝑥 ) = 𝑥 s i n .

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

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

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

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

What is the radius 𝑅 of convergence of the Maclaurin series for s i n 𝑥 ?

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

Q3:

Consider the function 𝑓 ( 𝑥 ) = 𝑥 c o s .

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

Write the general form for the 𝑛 th 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 2 ( ) c o s
  • E 𝑓 ( 𝑥 ) = ( 𝑥 + 𝑛 𝜋 ) ( ) c o s

Hence, derive the Maclaurin series for c o s 𝑥 .

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

What is the radius 𝑅 of convergence of the Maclaurin series for c o s 𝑥 ?

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

Q4:

Find the Maclaurin series of c o s h 2 𝑥 = 𝑒 + 𝑒 2 .

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

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