Worksheet: Differentiating and Integrating Power Series

In this worksheet, we will practice differentiating and integrating a power series using term-by-term differentiation and integration and using the results to find power series representation of some functions.

Q1:

For the given function 𝑓(𝑥)=(2𝑥)tan, find a power series representation for 𝑓 by integrating the power series for 𝑓.

  • A(1)𝑥2𝑛+1
  • B(1)𝑥2𝑛1
  • C2(1)𝑥
  • D2(1)𝑥2𝑛+1
  • E2(1)𝑥2𝑛+1

Q2:

Consider the series 𝑓(𝑥)=11+𝑥=(1)𝑥.

Differentiate the given series expansion of 𝑓 term by term to find the corresponding series expansion for the derivative of 𝑓.

  • A𝑛(1)𝑥
  • B𝑛(1)𝑥
  • C(1)(𝑛+1)𝑥
  • D(1)𝑥
  • E(1)𝑥

Use the result of the first part to evaluate the sum of the series (1)(𝑛+1)3.

  • A34
  • B43
  • C169
  • D916
  • E916

Q3:

Consider the series 𝑓(𝑥)=1(1𝑥)=(𝑛+1)𝑥.

Differentiate the given series expansion of 𝑓 term by term to find the corresponding series expansion for the derivative of 𝑓.

  • A𝑛(𝑛+1)𝑥
  • B(𝑛+1)𝑥
  • C(𝑛+1)(𝑛+2)𝑥
  • D(𝑛+1)(𝑛+2)𝑥
  • E𝑛(𝑛+1)𝑥

Use the result of the first part to evaluate the sum of the series (𝑛+1)(𝑛+2)4.

  • A12827
  • B227
  • C12827
  • D43
  • E43

Q4:

Consider the series 𝑓(𝑥)=11𝑥=𝑥.

Differentiate the given series expansion of 𝑓 term by term to find the corresponding series expansion for the derivative of 𝑓.

  • A3(𝑛+1)𝑥
  • B3𝑛𝑥
  • C𝑥
  • D3(𝑛+1)𝑥
  • E()(3𝑛1)𝑥

Use the result of the first part to evaluate the sum of the series 12(𝑛+1)2.

  • A17
  • B4912
  • C1249
  • D1249
  • E17

Q5:

For the given function 𝑓(𝑥)=(1+2𝑥)ln, find a power series representation for 𝑓 by integrating the power series for 𝑓.

  • A2(1)(𝑛+1)𝑥
  • B2(1)𝑥𝑛+1
  • C2(1)𝑥𝑛+2
  • D2(1)𝑥𝑛+1
  • E2(1)(𝑛+1)𝑥

Q6:

For the given function 𝑓(𝑥)=(3𝑥)tan, find the interval of convergence of the power series representation of 𝑓 by integrating the power series of 𝑓.

  • A13,13
  • B13,13
  • C13,13
  • D19,19
  • E19,19

Q7:

For the given function 𝑓(𝑥)=1+𝑥ln, find a power series representation of 𝑓 by integrating the power series of 𝑓.

  • A2(1)𝑥𝑛+1
  • B(1)𝑥𝑛+2
  • C(1)𝑥2(𝑛+1)
  • D(1)𝑥𝑛+1
  • E(1)𝑥2(𝑛+2)

Q8:

Consider the series 𝑓(𝑥)=11𝑥=𝑥. Find the interval of convergence for the derivative of the given series.

  • A(,)
  • B(1,1]
  • C(1,)
  • D(1,1)
  • E[1,1)

Q9:

Consider the series 𝑓(𝑥)=12𝑥(1𝑥)=𝑥2. Differentiate the given series expansion of 𝑓 term by term to find the corresponding series expansion for the derivative of 𝑓.

  • A(3𝑛+2)𝑥
  • B(3𝑛+2)2𝑥
  • C(3𝑛1)𝑥
  • D(3𝑛1)2𝑥
  • E(3𝑛)2𝑥

Q10:

Consider the series 𝑓(𝑥)=𝑥1𝑥=𝑥. Differentiate the given series expansion of 𝑓 term by term to find the corresponding series expansion for the derivative of 𝑓.

  • A(2𝑛+3)𝑥
  • B𝑥
  • C2𝑛𝑥
  • D(2𝑛+1)𝑥
  • E(2𝑛+2)𝑥

Q11:

Find 2𝑦7𝑦+5𝑦 given 𝑦=1+𝑥1!+𝑥2!+𝑥3!+.

Q12:

Consider the function 𝑓(𝑥)=152𝑥5tan. Find a power series expansion for 𝑓 by integrating the power series of 𝑓.

  • A6425𝑥2𝑛
  • B6425𝑥2𝑛+1
  • C6425𝑥2𝑛+1
  • D15425𝑥2𝑛+1
  • E6425𝑥2𝑛1

Q13:

Consider the function 𝑓(𝑥)=6𝑥𝑥+4𝑥+4. Find a power series expansion for 𝑓 by differentiating the power series of 𝐹.

  • A32(𝑛+1)𝑥
  • B(3)12(𝑛)𝑥
  • C(3)12(𝑛+1)𝑥
  • D(3)12(𝑛)𝑥
  • E32(𝑛+1)𝑥

Q14:

Consider the function 𝑓(𝑥)=18(13𝑥). Find a power series expansion for 𝑓 by differentiating the power series of 𝐹.

  • A63(𝑛1)𝑥
  • B18((𝑛+1)𝑥)
  • C183(𝑛+1)𝑥
  • D63(𝑛+1)𝑥
  • E63(𝑛1)𝑥

Q15:

Consider the function 𝑓(𝑥)=20𝑥𝑒. Find a power series expansion for 𝑓 by differentiating the power series of 𝐹.

  • A20(𝑛+1)!(7)(𝑥)
  • B10(𝑛+1)!(14)(𝑥)
  • C20𝑛!(7)(𝑥)
  • D14𝑛!(7)(𝑥)
  • E14(𝑛1)!(14)(𝑥)

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