Worksheet: Maclaurin and Taylor Series of Common Functions

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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)=(2−𝑥)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=𝑒−𝑒2.

  • 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 ln1−𝑥2.

  • 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 (𝑥−1).

  • 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 (𝑥−𝜋).

  • 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 𝑥−𝜋2, 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 (𝑥−𝜋).

  • 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 )

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