Worksheet: Taylor Series

In this worksheet, we will practice finding Taylor series of a function and finding the radius of convergence of the series.

Q1:

The following table shows the value of function 𝑓 and some of its derivatives at đ‘Ĩ=−2.

𝑓(−2)𝑓(−2)īŽ˜đ‘“(−2)(īŠ¨)𝑓(−2)(īŠŠ)𝑓(−2)(īŠĒ)
3−158−6

Write the first 5 terms of the Taylor series of 𝑓.

  • A3−(đ‘Ĩ+2)+5(đ‘Ĩ+2)+8(đ‘Ĩ+2)−6(đ‘Ĩ+2)īŠ¨īŠŠīŠĒ
  • B3−(đ‘Ĩ+2)+52(đ‘Ĩ+2)+43(đ‘Ĩ+2)−14(đ‘Ĩ+2)īŠ¨īŠŠīŠĒ
  • C3−(đ‘Ĩ−2)+5(đ‘Ĩ−2)+8(đ‘Ĩ−2)−6(đ‘Ĩ+2)īŠ¨īŠŠīŠĒ
  • D3−(đ‘Ĩ−2)+52(đ‘Ĩ−2)+43(đ‘Ĩ−2)−14(đ‘Ĩ−2)īŠ¨īŠŠīŠĒ
  • E3−(đ‘Ĩ+2)+52(đ‘Ĩ+2)+83(đ‘Ĩ+2)−32(đ‘Ĩ+2)īŠ¨īŠŠīŠĒ

Q2:

For the function 𝑓: 𝑓(3)=2, 𝑓(3)=7īŽ˜ and 𝑓(3)=−12𝑛𝑓(3)(īŠ)(īŠīŠąīŠ§) for 𝑛â‰Ĩ2.

Find the first five terms of the Taylor series representation of 𝑓 at đ‘Ĩ=3.

  • A2+7(đ‘Ĩ−3)−12(đ‘Ĩ−3)−14(đ‘Ĩ−3)−112(đ‘Ĩ−3)īŠ¨īŠŠīŠĒ
  • B2+7(đ‘Ĩ−3)−74(đ‘Ĩ−3)+724(đ‘Ĩ−3)−7192(đ‘Ĩ−3)īŠ¨īŠŠīŠĒ
  • C2+7(đ‘Ĩ−3)−7(đ‘Ĩ−3)+212(đ‘Ĩ−3)−21(đ‘Ĩ−3)īŠ¨īŠŠīŠĒ
  • D2+7(đ‘Ĩ−3)−72(đ‘Ĩ−3)+74(đ‘Ĩ−3)−78(đ‘Ĩ−3)īŠ¨īŠŠīŠĒ
  • E2+7(đ‘Ĩ−3)−72(đ‘Ĩ−3)+78(đ‘Ĩ−3)−796(đ‘Ĩ−3)īŠ¨īŠŠīŠĒ

Q3:

Consider the function 𝑓(đ‘Ĩ)=(2+đ‘Ĩ)īŠ§īŠ¨.

Find the Taylor series representation of the function 𝑓(đ‘Ĩ)=(2+đ‘Ĩ)īŠ§īŠ¨ at đ‘Ĩ=−1.

  • A−1+12(đ‘Ĩ+1)−(12)(11)(đ‘Ĩ+1)2+(12)(11)(10)(đ‘Ĩ+1)6−(12)(11)(10)(9)(đ‘Ĩ+1)24+⋯īŠ¨īŠŠīŠĒ
  • B1+(đ‘Ĩ+1)+(đ‘Ĩ+1)2+(đ‘Ĩ+1)6+(đ‘Ĩ+1)24+⋯īŠ¨īŠŠīŠĒ
  • C1+12(2)(đ‘Ĩ+1)+(12)(11)(2)(đ‘Ĩ+1)2+(12)(11)(10)(2)(đ‘Ĩ+1)6+(12)(11)(10)(9)(2)(đ‘Ĩ+1)24+⋯īŠ¨īŠ¨īŠŠīŠŠīŠĒīŠĒ
  • D−1+(đ‘Ĩ+1)−(đ‘Ĩ+1)2+(đ‘Ĩ+1)6−(đ‘Ĩ+1)24+⋯īŠ¨īŠŠīŠĒ
  • E1+12(đ‘Ĩ+1)+(12)(11)(đ‘Ĩ+1)2+(12)(11)(10)(đ‘Ĩ+1)6+(12)(11)(10)(9)(đ‘Ĩ+1)24+⋯īŠ¨īŠŠīŠĒ

Are the terms of the Taylor series representation for the function 𝑓 finite or infinite?

  • AFinite
  • BInfinite

Q4:

Consider the function 𝑓(đ‘Ĩ)=(đ‘Ĩ)cos.

Find the Taylor series expansion of 𝑓(đ‘Ĩ)=(đ‘Ĩ)cos at đ‘Ĩ=𝜋.

  • A−(đ‘Ĩ−𝜋)−(đ‘Ĩ−𝜋)6−(đ‘Ĩ−𝜋)120−⋯īŠŠīŠĢ
  • B−1−(đ‘Ĩ−𝜋)2−(đ‘Ĩ−𝜋)24−(đ‘Ĩ−𝜋)720−⋯īŠ¨īŠĒīŠŦ
  • C−(đ‘Ĩ−𝜋)+(đ‘Ĩ−𝜋)6−(đ‘Ĩ−𝜋)120+⋯īŠŠīŠĢ
  • D−1+(đ‘Ĩ−𝜋)2−(đ‘Ĩ−𝜋)24+(đ‘Ĩ−𝜋)720−⋯īŠ¨īŠĒīŠŦ
  • E−1+(đ‘Ĩ−𝜋)−(đ‘Ĩ−𝜋)2+(đ‘Ĩ−𝜋)6−(đ‘Ĩ−𝜋)24+⋯īŠ¨īŠŠīŠĒ

Write the Taylor series expansion of 𝑓(đ‘Ĩ) in sigma notation.

  • A∞ī‰īŠ˛īŠĻī‰īŠ°īŠ§īŠ¨ī‰īŠ°īŠ§ī„š(−1)(đ‘Ĩ−𝜋)(2𝑚+1)!
  • B∞ī‰īŠ˛īŠĻīŠ¨ī‰ī„šâˆ’(đ‘Ĩ−𝜋)(2𝑚)!
  • C∞ī‰īŠ˛īŠĻī‰īŠ°īŠ§ī‰ī„š(−1)(đ‘Ĩ−𝜋)𝑚!
  • D∞ī‰īŠ˛īŠĻīŠ¨ī‰īŠ°īŠ§ī„šâˆ’(đ‘Ĩ−𝜋)(2𝑚+1)!
  • E∞ī‰īŠ˛īŠĻī‰īŠ°īŠ§īŠ¨ī‰ī„š(−1)(đ‘Ĩ−𝜋)(2𝑚)!

Q5:

For a function 𝑓: 𝑓(−4)=6, 𝑓(−4)=−6īŽ˜ and 𝑓(−4)=−1𝑛𝑓(−4)(īŠ)(īŠīŠąīŠ§) for 𝑛â‰Ĩ2. Find the Taylor series expansion of 𝑓 at đ‘Ĩ=−4.

  • A∞īŠīŠ˛īŠĻīŠīŠī„š(−1)6(𝑛)(𝑛!)(đ‘Ĩ+4)
  • B∞īŠīŠ˛īŠĻīŠīŠ°īŠ§īŠī„š(−1)6(𝑛!)(đ‘Ĩ+4)
  • C∞īŠīŠ˛īŠĻīŠīŠ¨īŠī„š(−1)6(𝑛!)(đ‘Ĩ+4)
  • D∞īŠīŠ˛īŠĻīŠīŠī„š(−1)6(𝑛!)(đ‘Ĩ+4)
  • E∞īŠīŠ˛īŠĻīŠīŠ°īŠ§īŠ¨īŠī„š(−1)6(𝑛!)(đ‘Ĩ+4)

Q6:

The Taylor series representation of the function 𝑓 is given by 𝑓(đ‘Ĩ)=2−4(đ‘Ĩ−7)+3(đ‘Ĩ−7)−(đ‘Ĩ−7)6+⋯īŠ¨īŠĒīŠŦ.

Find the value of the fourth derivative of 𝑓 at đ‘Ĩ=7.

  • A18
  • B3
  • C72
  • D0
  • E6

Find the value of the third derivative of 𝑓 at đ‘Ĩ=7.

Q7:

What is the necessary and sufficient condition for the function 𝑓 to have a Taylor series representation about đ‘Ĩ=𝑎?

  • A𝑓(𝑎)(īŠ) is not equal to 𝑓(𝑎) for any positive integer 𝑛. (𝑓(īŠ) is the 𝑛th derivative of 𝑓.)
  • B𝑓(𝑎) exists.
  • C𝑓(𝑎) is not equal to 0.
  • D𝑓(𝑎)(īŠ) is not equal to 0 for any positive integer 𝑛. (𝑓(īŠ) is the 𝑛th derivative of 𝑓.)
  • E𝑓(𝑎)(īŠ) exists for any positive integer 𝑛. (𝑓(īŠ) is the 𝑛th derivative of 𝑓.)

Q8:

Consider the function 𝑓(đ‘Ĩ)=(đ‘Ĩ)ln.

Find the Taylor series representation of 𝑓 about đ‘Ĩ=1.

  • A∞īŠīŠ˛īŠ§īŠīŠ°īŠ§īŠī„š(−1)𝑛(đ‘Ĩ−1)𝑛!
  • B∞īŠīŠ˛īŠ§īŠīŠ°īŠ§īŠī„š(−1)(𝑛−1)(đ‘Ĩ−1)𝑛!
  • C∞īŠīŠ˛īŠ§īŠīŠ°īŠ§īŠī„š(−1)(đ‘Ĩ−1)𝑛
  • D∞īŠīŠ˛īŠĻīŠīŠ°īŠ§īŠī„š(−1)(đ‘Ĩ−1)𝑛
  • E∞īŠīŠ˛īŠĻīŠīŠ°īŠ§īŠī„š(−1)(𝑛−1)(đ‘Ĩ−1)𝑛!

Find the interval of convergence of the Taylor series representation of 𝑓 about đ‘Ĩ=1.

  • A[0,2]
  • B(0,2]
  • C(−∞,∞)
  • D(−1,1]
  • E[−1,1]

What is the radius of convergence of the Taylor series representation of 𝑓 about đ‘Ĩ=1?

  • A2
  • B∞
  • C0
  • D12
  • E1

Q9:

What are the first four terms of the Taylor series of the function 𝑓(đ‘Ĩ)=√đ‘Ĩ about đ‘Ĩ=4?

  • A𝑓(đ‘Ĩ)=2+12(đ‘Ĩ−4)−18(đ‘Ĩ−4)+116(đ‘Ĩ−4)−⋯īŠ¨īŠŠ
  • B𝑓(đ‘Ĩ)=2+14(đ‘Ĩ−4)−164(đ‘Ĩ−4)+1512(đ‘Ĩ−4)−⋯īŠ¨īŠŠ
  • C𝑓(đ‘Ĩ)=2+12(đ‘Ĩ−4)−196(đ‘Ĩ−4)+1192(đ‘Ĩ−4)−⋯īŠ¨īŠŠ
  • D𝑓(đ‘Ĩ)=2+12(đ‘Ĩ−4)−116(đ‘Ĩ−4)+1192(đ‘Ĩ−4)−⋯īŠ¨īŠŠ
  • E𝑓(đ‘Ĩ)=2+14(đ‘Ĩ−4)−132(đ‘Ĩ−4)+3256(đ‘Ĩ−4)−⋯īŠ¨īŠŠ

Q10:

Consider the function 𝑓(đ‘Ĩ)=𝑒īŠ¨ī—.

Find the Taylor series representation of 𝑓 about đ‘Ĩ=3.

  • A∞īŠīŠ˛īŠĻīŠŦīŠī„šđ‘’(đ‘Ĩ−3)𝑛!
  • B∞īŠīŠ˛īŠĻīŠī„š(2đ‘Ĩ−3)𝑛!
  • C∞īŠīŠ˛īŠĻīŠīŠŦīŠī„š2𝑒(đ‘Ĩ−3)𝑛!
  • D∞īŠīŠ˛īŠĻīŠŠīŠī„šđ‘’(2đ‘Ĩ−3)𝑛!
  • E∞īŠīŠ˛īŠĻīŠīŠī„š2(đ‘Ĩ)𝑛!

Find the interval of convergence of the Taylor series representation of 𝑓 about đ‘Ĩ=3.

  • Aī”52,72īˆ
  • Bī€ŧ52,72īˆ
  • Cī”52,72ī 
  • Dī€ŧ52,72ī 
  • E(−∞,∞)

What is the radius of convergence of the Taylor series representation of 𝑓 about đ‘Ĩ=3?

  • A∞
  • B2
  • C12
  • D0
  • E1

Q11:

Find the Taylor series of the function 𝑓(đ‘Ĩ)=(1−đ‘Ĩ)īŠąīŠ¨ about đ‘Ĩ=−5.

  • A∞īŠīŠ˛īŠĻīŠąīŠ¨īŠąīŠīŠī„š12(𝑛+1)(đ‘Ĩ+5)
  • B∞īŠīŠ˛īŠĻīŠąīŠ¨īŠąīŠīŠī„š6(𝑛+1)(đ‘Ĩ−5)
  • C∞īŠīŠ˛īŠĻīŠąīŠ¨īŠąīŠīŠī„š6(𝑛+1)(đ‘Ĩ+5)
  • D∞īŠīŠ˛īŠĻīŠąīŠ¨īŠąīŠīŠī„š(−6)(𝑛+1)(đ‘Ĩ−5)
  • E∞īŠīŠ˛īŠĻīŠąīŠ¨īŠąīŠīŠī„š(−6)(𝑛+1)(đ‘Ĩ+5)

Q12:

Write the first four terms of the Taylor series expansion of 𝑓(đ‘Ĩ)=11𝑒īŠ¨ī— in ascending powers of (đ‘Ĩ−2).

  • A−11𝑒−22𝑒(đ‘Ĩ−2)−22𝑒(đ‘Ĩ−2)−443𝑒(đ‘Ĩ−2)īŠĒīŠĒīŠĒīŠ¨īŠĒīŠŠ
  • B22𝑒+44𝑒(đ‘Ĩ−2)+44𝑒(đ‘Ĩ−2)+443𝑒(đ‘Ĩ−2)īŠĒīŠĒīŠĒīŠ¨īŠĒīŠŠ
  • C11𝑒+22𝑒(đ‘Ĩ−2)+22𝑒(đ‘Ĩ−2)+443𝑒(đ‘Ĩ−2)īŠ¨īŠ¨īŠ¨īŠ¨īŠ¨īŠŠ
  • D11𝑒+22𝑒(đ‘Ĩ−2)+22𝑒(đ‘Ĩ−2)+443𝑒(đ‘Ĩ−2)īŠĒīŠĒīŠĒīŠ¨īŠĒīŠŠ
  • E−11𝑒−22𝑒(đ‘Ĩ−2)−22𝑒(đ‘Ĩ−2)−443𝑒(đ‘Ĩ−2)īŠ¨īŠ¨īŠ¨īŠ¨īŠ¨īŠŠ

Q13:

Write the first three terms of the Taylor series expansion of 𝑓(đ‘Ĩ)=1√1+2đ‘Ĩ in ascending powers of (đ‘Ĩ−1).

  • A−2√3+2(đ‘Ĩ−1)3√3−2(đ‘Ĩ−1)6√3īŠ¨
  • B12√3−(đ‘Ĩ−1)6√3+(đ‘Ĩ−1)12√3īŠ¨
  • C1√3−(đ‘Ĩ−1)3√3+(đ‘Ĩ−1)6√3īŠ¨
  • D−1√3+(đ‘Ĩ−1)3√3−(đ‘Ĩ−1)6√3īŠ¨
  • E2√3−2(đ‘Ĩ−1)3√3+2(đ‘Ĩ−1)6√3īŠ¨

Q14:

Find the radius of convergence for the Taylor series of 𝑓(đ‘Ĩ)=2đ‘Ĩsin about đ‘Ĩ=𝜋.

  • A𝜋
  • B1
  • C∞
  • D12
  • E𝜋2

Q15:

Find the radius of convergence for the Taylor series of 𝑓(đ‘Ĩ)=12đ‘Ĩ+1 about đ‘Ĩ=1.

  • A34
  • B32
  • C2
  • D23
  • E12

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