Worksheet: Maclaurin and Taylor Series of Common Functions

In this worksheet, we will practice finding the Taylor/Maclaurin series representation of common functions such as exponential and trigonometric functions and binomial expansion.

Q1:

Consider 𝑓(𝑥)=(2𝑥)ln.

Find a power series representation for 𝑓(𝑥).

  • A 𝑓 ( 𝑥 ) = ( 2 ) + 𝑥 2 ( 𝑛 + 1 ) l n
  • B 𝑓 ( 𝑥 ) = ( 2 ) + 𝑥 2 l n
  • C 𝑓 ( 𝑥 ) = ( 2 ) 𝑥 2 ( 𝑛 + 1 ) l n
  • D 𝑓 ( 𝑥 ) = ( 2 ) 𝑥 2 l n
  • E 𝑓 ( 𝑥 ) = ( 2 ) 1 𝑛 𝑥 2 l n

Find its interval of convergence.

  • A | 𝑥 | < 2
  • B | 𝑥 | < 1
  • C | 𝑥 | > 1
  • D | 𝑥 | > 0
  • E | 𝑥 | > 2

Q2:

Consider 𝑔(𝑥)=𝑒.

Find the Maclaurin series of 𝑔(𝑥).

  • A 𝑔 ( 𝑥 ) = 𝑥 𝑛 !
  • B 𝑔 ( 𝑥 ) = 𝑥 𝑛 !
  • C 𝑔 ( 𝑥 ) = 𝑥 𝑛 !
  • D 𝑔 ( 𝑥 ) = 𝑥 ( 𝑛 + 1 ) !
  • E 𝑔 ( 𝑥 ) = 𝑥 𝑛 !

Use the first three terms of this series to find an approximate value of 𝑒 to 2 decimal places.

Q3:

The function cos𝑥 can be represented by the power series (1)(2𝑛)!𝑥. Use the first two terms of this series to find an approximate value of cos0.5 to two decimal places.

Q4:

The function sin𝑥 can be represented by the power series (1)(2𝑛+1)!𝑥. Use the first two terms of this series to find an approximate value of sin0.5 to 2 decimal places.

Q5:

Consider the binomial expansion for 1+1𝑛.

Which of the following expressions is its fourth term?

  • A 1 1 3 ! ( 𝑛 3 ) !
  • B 𝑛 ! 3 !
  • C 1 1 3 !
  • D 1 1 + 3 !
  • E 1 1 𝑛 1 2 𝑛

What is the limit of the (𝑘+1)th term as 𝑛 tends to infinity?

  • A 1 𝑘 ! ( 𝑘 1 ) !
  • B 1 𝑘 !
  • C
  • D 1 ( 𝑘 + 1 ) !
  • E1

Hence, write in summation (or sigma) notation a series which is equal to the limit of 1+1𝑛 as 𝑛 tends to infinity.

  • A 1 𝑛 ! ( 𝑛 1 ) !
  • B ( 𝑛 2 ) ! ( 𝑛 1 ) !
  • C 1 𝑛 !

What is the value of this series?

  • A 𝑒
  • B 𝑖
  • C 𝜑
  • D 𝜋

Q6:

Find the Maclaurin series of sinh3𝑥=𝑒𝑒2.

  • A ( 𝑥 ) ( 2 𝑛 + 1 ) !
  • B ( 1 ) ( 𝑥 ) ( 2 𝑛 + 1 ) !
  • C ( 1 ) ( 3 𝑥 ) ( 2 𝑛 ) !
  • D ( 3 𝑥 ) ( 2 𝑛 ) !
  • E ( 3 𝑥 ) ( 2 𝑛 + 1 ) !

Q7:

Use the Maclaurin series of sin𝑥 to express 𝑥𝑥sind as an infinite series.

  • A ( 1 ) 𝑥 ( 2 𝑛 + 1 ) ! ( 6 𝑛 + 4 ) + 𝑐
  • B ( 1 ) 𝑥 ( 2 𝑛 + 1 ) ! + 𝑐
  • C 𝑥 ( 6 𝑛 + 4 ) ! + 𝑐
  • D ( 1 ) 𝑥 ( 6 𝑛 + 4 ) ! + 𝑐
  • E 𝑥 ( 2 𝑛 + 1 ) ! ( 6 𝑛 + 4 ) + 𝑐

Q8:

Use the Maclaurin series of 𝑒 to express 𝑒𝑥d as an infinite series.

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

Q9:

Find the Maclaurin series of ln1𝑥2.

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

Q10:

Write the first three terms of the Taylor expansion for 𝑓(𝑥)=𝑒 about 1 in ascending powers of (𝑥1).

  • A 𝑒 + 2 𝑒 ( 𝑥 1 ) + 3 𝑒 ( 𝑥 1 )
  • B 𝑒 + 2 𝑒 ( 𝑥 + 1 ) + 3 𝑒 ( 𝑥 + 1 )
  • C 𝑒 2 𝑒 ( 𝑥 1 ) 3 𝑒 ( 𝑥 1 )
  • D 𝑒 + 2 𝑒 ( 𝑥 1 ) + 2 𝑒 ( 𝑥 1 )
  • E 𝑒 + 2 𝑒 ( 𝑥 1 ) + 3 𝑒 ( 𝑥 1 )

Q11:

Write the first three terms of the Taylor expansion for 𝑓(𝑥)=𝑥cos about 𝜋 in ascending powers of (𝑥𝜋).

  • A 1 2 1 4 ( 𝑥 𝜋 ) + 1 4 8 ( 𝑥 𝜋 )
  • B 1 + 1 2 ( 𝑥 𝜋 ) 1 2 4 ( 𝑥 𝜋 )
  • C 1 + 1 2 ( 𝑥 𝜋 ) 1 2 4 ( 𝑥 𝜋 )
  • D 1 2 + 1 4 ( 𝑥 𝜋 ) 1 4 8 ( 𝑥 𝜋 )
  • E 1 1 2 ( 𝑥 𝜋 ) + 1 2 4 ( 𝑥 𝜋 )

Q12:

By writing the first three nonzero terms of the Taylor expansion of 𝑓(𝑥)=2𝑥sin about 𝜋2 in ascending powers of 𝑥𝜋2, estimate the value of sin1.6. Give your answer accurate to three significant figures.

Q13:

By writing the first three nonzero terms of the Maclaurin expansion of 𝑓(𝑥)=𝑥tan in ascending powers of 𝑥, estimate the value of tan𝜋4. Give your answer accurate to three significant figures.

Q14:

Find the first three nonzero terms of the Taylor expansion for 𝑓(𝑥)=2𝑒𝑥sin about 𝜋, in ascending powers of (𝑥𝜋).

  • A 2 𝑒 ( 𝑥 𝜋 ) 2 𝑒 ( 𝑥 𝜋 ) 1 3 𝑒 ( 𝑥 𝜋 )
  • B 2 𝑒 ( 𝑥 𝜋 ) 2 𝑒 ( 𝑥 𝜋 ) 2 3 𝑒 ( 𝑥 𝜋 )
  • C 2 𝑒 ( 𝑥 𝜋 ) + 2 𝑒 ( 𝑥 𝜋 ) + 2 3 𝑒 ( 𝑥 𝜋 )
  • D 2 𝑒 ( 𝑥 + 𝜋 ) 2 𝑒 ( 𝑥 + 𝜋 ) 2 3 𝑒 ( 𝑥 + 𝜋 )
  • E 2 𝑒 ( 𝑥 𝜋 ) 2 𝑒 ( 𝑥 𝜋 ) 2 3 𝑒 ( 𝑥 𝜋 )

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.