Lesson Worksheet: Maclaurin and Taylor Series of Common Functions Mathematics • Higher Education

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 ๐‘“(๐‘ฅ)=(2โˆ’๐‘ฅ)ln.

Find a power series representation for ๐‘“(๐‘ฅ).

  • A๐‘“(๐‘ฅ)=(2)+๏„š๐‘ฅ2(๐‘›+1)lnโˆž๏Š๏Šฒ๏Šง๏Š๏Šฐ๏Šง๏Š๏Šฐ๏Šง
  • B๐‘“(๐‘ฅ)=(2)+๏„š๏€ป๐‘ฅ2๏‡lnโˆž๏Š๏Šฒ๏Šฆ๏Š๏Šฐ๏Šง
  • C๐‘“(๐‘ฅ)=(2)โˆ’๏„š๐‘ฅ2(๐‘›+1)lnโˆž๏Š๏Šฒ๏Šฆ๏Š๏Šฐ๏Šง๏Š๏Šฐ๏Šง
  • D๐‘“(๐‘ฅ)=(2)โˆ’๏„š๏€ป๐‘ฅ2๏‡lnโˆž๏Š๏Šฒ๏Šง๏Š๏Šฐ๏Šง
  • E๐‘“(๐‘ฅ)=(2)โˆ’๏„š๏€ผ1๐‘›๏ˆ๏€ป๐‘ฅ2๏‡lnโˆž๏Š๏Šฒ๏Šฆ๏Š

Find its interval of convergence.

  • A|๐‘ฅ|<2
  • B|๐‘ฅ|<1
  • C|๐‘ฅ|>1
  • D|๐‘ฅ|>0
  • E|๐‘ฅ|>2

Q2:

Consider ๐‘”(๐‘ฅ)=๐‘’๏—.

Find the Maclaurin series of ๐‘”(๐‘ฅ).

  • A๐‘”(๐‘ฅ)=๏„š๐‘ฅ๐‘›!โˆž๏Š๏Šฒ๏Šฆ๏Š๏Šฐ๏Šง
  • B๐‘”(๐‘ฅ)=๏„š๐‘ฅ๐‘›!โˆž๏Š๏Šฒ๏Šฆ๏Š
  • C๐‘”(๐‘ฅ)=๏„š๐‘ฅ๐‘›!โˆž๏Š๏Šฒ๏Šง๏Š๏Šฐ๏Šง
  • D๐‘”(๐‘ฅ)=๏„š๐‘ฅ(๐‘›+1)!โˆž๏Š๏Šฒ๏Šฆ๏Š๏Šฐ๏Šง
  • E๐‘”(๐‘ฅ)=๏„š๐‘ฅ๐‘›!โˆž๏Š๏Šฒ๏Šง๏Š

Use the first three terms of this series to find an approximate value of ๐‘’๏Šฆ๏Ž–๏Šช to 2 decimal places.

Q3:

The function cos๐‘ฅ can be represented by the power series โˆž๏Š๏Šฒ๏Šฆ๏Š๏Šจ๏Š๏„š(โˆ’1)(2๐‘›)!๐‘ฅ. Use the first two terms of this series to find an approximate value of cos0.5 to two decimal places.

Q4:

The function sin๐‘ฅ can be represented by the power series โˆž๏Š๏Šฒ๏Šฆ๏Š๏Šจ๏Š๏Šฐ๏Šง๏„š(โˆ’1)(2๐‘›+1)!๐‘ฅ. 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 ๏€ผ1+1๐‘›๏ˆ๏Š.

Which of the following expressions is its fourth term?

  • A๏€ป1โˆ’๏‡๏€ป1โˆ’๏‡3!(๐‘›โˆ’3)!๏Šง๏Š๏Šจ๏Š
  • B๐‘›!3!
  • C๏€ป1โˆ’๏‡๏€ป1โˆ’๏‡3!๏Šง๏Š๏Šจ๏Š
  • D๏€ป1โˆ’๏‡๏€ป1+๏‡3!๏Šง๏Š๏Šจ๏Š
  • E๏€ผ1โˆ’1๐‘›๏ˆ๏€ผ1โˆ’2๐‘›๏ˆ

What is the limit of the (๐‘˜+1)th term as ๐‘› tends to infinity?

  • A1๐‘˜!(๐‘˜โˆ’1)!
  • B1๐‘˜!
  • Cโˆž
  • D1(๐‘˜+1)!
  • E1

Hence, write in summation (or sigma) notation a series which is equal to the limit of ๏€ผ1+1๐‘›๏ˆ๏Š as ๐‘› tends to infinity.

  • Aโˆž๏Š๏Šฒ๏Šฆ๏„š1๐‘›!(๐‘›โˆ’1)!
  • Bโˆž๏Š๏Šฒ๏Šฆ๏„š(๐‘›โˆ’2)!(๐‘›โˆ’1)!
  • Cโˆž๏Š๏Šฒ๏Šฆ๏„š1๐‘›!

What is the value of this series?

  • A๐‘’
  • B๐‘–
  • C๐œ‘
  • D๐œ‹

Q6:

Find the Maclaurin series of sinh3๐‘ฅ=๐‘’โˆ’๐‘’2๏Šฉ๏—๏Šฑ๏Šฉ๏—.

  • Aโˆž๏Š๏Šฒ๏Šฆ๏Šจ๏Š๏Šฐ๏Šง๏„š(๐‘ฅ)(2๐‘›+1)!
  • Bโˆž๏Š๏Šฒ๏Šฆ๏Š๏Šจ๏Š๏Šฐ๏Šง๏„š(โˆ’1)(๐‘ฅ)(2๐‘›+1)!
  • Cโˆž๏Š๏Šฒ๏Šฆ๏Š๏Šจ๏Š๏„š(โˆ’1)(3๐‘ฅ)(2๐‘›)!
  • Dโˆž๏Š๏Šฒ๏Šฆ๏Šจ๏Š๏„š(3๐‘ฅ)(2๐‘›)!
  • Eโˆž๏Š๏Šฒ๏Šฆ๏Šจ๏Š๏Šฐ๏Šง๏„š(3๐‘ฅ)(2๐‘›+1)!

Q7:

Use the Maclaurin series of ๐‘’๏— to express ๏„ธ๐‘’๐‘ฅ๏—๏Žกd as an infinite series.

  • Aโˆž๏Š๏Šฒ๏Šฆ๏Šจ๏Š๏Šฐ๏Šง๏„š๐‘ฅ๐‘›!+๐‘
  • Bโˆž๏Š๏Šฒ๏Šฆ๏Šจ๏Š๏Šฐ๏Šง๏„š๐‘ฅ(2๐‘›+1)!+๐‘
  • Cโˆž๏Š๏Šฒ๏Šฆ๏Šจ๏Š๏„š๐‘ฅ๐‘›!+๐‘
  • Dโˆž๏Š๏Šฒ๏Šฆ๏Šจ๏Š๏„š๐‘ฅ(2๐‘›)!+๐‘
  • Eโˆž๏Š๏Šฒ๏Šฆ๏Šจ๏Š๏Šฐ๏Šง๏„š๐‘ฅ๐‘›!(2๐‘›+1)+๐‘

Q8:

Use the Maclaurin series of sin๐‘ฅ to express ๏„ธ๏€น๐‘ฅ๏…๐‘ฅsind๏Šฉ as an infinite series.

  • Aโˆž๏Š๏Šฒ๏Šฆ๏Š๏Šฌ๏Š๏Šฐ๏Šช๏„š(โˆ’1)๐‘ฅ(2๐‘›+1)!(6๐‘›+4)+๐‘
  • Bโˆž๏Š๏Šฒ๏Šฆ๏Š๏Šฌ๏Š๏Šฐ๏Šฉ๏„š(โˆ’1)๐‘ฅ(2๐‘›+1)!+๐‘
  • Cโˆž๏Š๏Šฒ๏Šฆ๏Šฌ๏Š๏Šฐ๏Šช๏„š๐‘ฅ(6๐‘›+4)!+๐‘
  • Dโˆž๏Š๏Šฒ๏Šฆ๏Š๏Šฌ๏Š๏Šฐ๏Šช๏„š(โˆ’1)๐‘ฅ(6๐‘›+4)!+๐‘
  • Eโˆž๏Š๏Šฒ๏Šฆ๏Šฌ๏Š๏Šฐ๏Šช๏„š๐‘ฅ(2๐‘›+1)!(6๐‘›+4)+๐‘

Q9:

Find the Maclaurin series of ln๏€ป1โˆ’๐‘ฅ2๏‡.

  • Aโˆ’๏„š1(๐‘›+1)!(๐‘ฅ)โˆž๏Š๏Šฒ๏Šฆ๏Š๏Šฐ๏Šง
  • Bโˆž๏Š๏Šฒ๏Šฆ๏Š๏Š๏Šฐ๏Šง๏„š(โˆ’1)1(๐‘›+1)๏€ป๐‘ฅ2๏‡
  • Cโˆ’๏„š1(๐‘›+1)๏€ป๐‘ฅ2๏‡โˆž๏Š๏Šฒ๏Šฆ๏Š๏Šฐ๏Šง
  • Dโˆž๏Š๏Šฒ๏Šฆ๏Š๏Š๏Šฐ๏Šง๏„š(โˆ’1)1(๐‘›+1)(๐‘ฅ)
  • Eโˆ’๏„š1(๐‘›+1)!๏€ป๐‘ฅ2๏‡โˆž๏Š๏Šฒ๏Šฆ๏Š๏Šฐ๏Šง

Q10:

Write the first three terms of the Taylor expansion for ๐‘“(๐‘ฅ)=๐‘ฅcos about ๐œ‹ in ascending powers of (๐‘ฅโˆ’๐œ‹).

  • A12โˆ’14(๐‘ฅโˆ’๐œ‹)+148(๐‘ฅโˆ’๐œ‹)๏Šจ๏Šช
  • Bโˆ’1+12(๐‘ฅโˆ’๐œ‹)โˆ’124(๐‘ฅโˆ’๐œ‹)๏Šช๏Šฎ
  • Cโˆ’1+12(๐‘ฅโˆ’๐œ‹)โˆ’124(๐‘ฅโˆ’๐œ‹)๏Šจ๏Šช
  • Dโˆ’12+14(๐‘ฅโˆ’๐œ‹)โˆ’148(๐‘ฅโˆ’๐œ‹)๏Šจ๏Šช
  • E1โˆ’12(๐‘ฅโˆ’๐œ‹)+124(๐‘ฅโˆ’๐œ‹)๏Šจ๏Šช

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