Worksheet: Power Series and Radius of Convergence

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

Q1:

Find the interval of convergence for the power series 𝑥 5 𝑛 + 1 0 .

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

Q2:

Find the interval of convergence of the power series 5 𝑥 3 .

  • A 2 7 2 5 , 2 7 2 5
  • B ( , )
  • C 3 5 , 3 5
  • D 3 5 , 3 5
  • E 2 7 2 5 , 2 7 2 5

Q3:

Determine the interval of convergence of the series ( 2 𝑥 + 5 ) ( 𝑛 + 1 ) 3 .

  • A [ 4 , 1 ]
  • B 1 2 , 1 2
  • C ( 4 , 1 )
  • D 3 2 , 3 2
  • E 3 2 , 3 2

Q4:

Consider the power series ( 1 ) ( 2 𝑥 + 9 ) 𝑛 + 6 .

Find the interval of convergence of the power series.

  • A ( 4 , 5 )
  • B [ 5 , 4 ]
  • C ( 5 , 4 ]
  • D ( 5 , 4 )
  • E [ 5 , 4 )

Find the radius of convergence of the power series.

Q5:

Find the radius of convergence for the power series ( 𝑛 1 ) ! ( 𝑥 5 ) 𝑛 .

Q6:

Consider the power series ( 3 𝑥 ) 𝑛 + 5 .

Find the interval of convergence of the power series.

  • A 1 3 , 1 3
  • B 1 3 , 1 3
  • C ( 1 , 1 )
  • D ( , )
  • E 1 3 , 1 3

Find the radius of convergence of the power series.

  • A1
  • B 2 3
  • C 1 3
  • D
  • E0

Q7:

Find the interval of convergence for the power series ( 𝑥 2 ) 𝑛 ! .

  • A [ 1 , 3 )
  • B [ 1 , 1 )
  • C ( 1 , 3 )
  • D ( 1 , 1 )
  • E ( , )

Q8:

Consider the power series 𝑛 ( 2 𝑥 ) .

Determine the radius of convergence of the power series.

  • A
  • B1
  • C0
  • D 1 2
  • E 2

Determine the interval of convergence of the power series.

  • A 1 2 , 1 2
  • BThe series converges only at 𝑥 = 0
  • C ( , )
  • D 1 2 , 1 2
  • E 1 2 , 1 2

Q9:

Consider the power series 𝑥 ( 4 𝑛 + 3 ) .

Determine the interval of convergence of the power series.

  • A 1 4 , 1 4
  • B ( 1 , 1 )
  • C 1 4 , 1 4
  • D [ 1 , 1 ]
  • E [ 1 , 1 )

Determine the radius of convergence of the power series.

Q10:

Consider the power series ( 1 ) ( 3 𝑥 + 4 ) 𝑛 .

Determine the radius of convergence of the power series.

  • A0
  • B 2 3
  • C
  • D1
  • E 1 3

Determine the interval of convergence of the power series.

  • A 5 3 , 1
  • B ( , )
  • C ( 5 , 3 )
  • D [ 5 , 3 ]
  • E 5 3 , 1

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