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 ) = ( 2 𝑥 ) t a n , find a power series representation for 𝑓 by integrating the power series for 𝑓 .

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

Q2:

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

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

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

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

  • A 9 1 6
  • B 4 3
  • C 1 6 9
  • D 9 1 6
  • E 3 4

Q3:

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

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

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

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

  • A 4 3
  • B 1 2 8 2 7
  • C 4 3
  • D 2 2 7
  • E 1 2 8 2 7

Q4:

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

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

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

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

  • A 1 2 4 9
  • B 4 9 1 2
  • C 1 7
  • D 1 2 4 9
  • E 1 7

Q5:

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

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

Q6:

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

  • A 1 9 , 1 9
  • B 1 3 , 1 3
  • C 1 3 , 1 3
  • D 1 9 , 1 9
  • E 1 3 , 1 3

Q7:

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

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

Q8:

Consider the series 𝑓 ( 𝑥 ) = 1 1 𝑥 = 𝑥 . Find the interval of convergence for the derivative of the given series.

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

Q9:

Consider the series 𝑓 ( 𝑥 ) = 1 2 𝑥 ( 1 𝑥 ) = 𝑥 2 . Differentiate the given series expansion of 𝑓 term by term to find the corresponding series expansion for the derivative of 𝑓 .

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

Q10:

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

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

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