Worksheet: Operations on Power Series

In this worksheet, we will practice adding, subtracting, and multiplying two power series and finding the radius of convergence of the resulting power series.

Q1:

Suppose that 𝑎 𝑥 is a power series whose interval of convergence is ( 3 , 3 ) and that 𝑏 𝑥 is a power series whose interval of convergence is ( 5 , 5 ) .

Find the interval of convergence of the series ( 𝑎 𝑥 𝑏 𝑥 ) .

  • A ( 3 , 5 )
  • B ( 5 , 5 )
  • C ( 5 , 3 )
  • D ( 8 , 8 )
  • E ( 3 , 3 )

Find the interval of convergence of the series 𝑏 2 𝑥 .

  • A ( 1 0 , 1 0 )
  • B ( 5 , 5 )
  • C 5 2 , 5 2
  • D ( 6 , 6 )
  • E ( 3 , 3 )

Q2:

Use partial fractions to find the power series of the function 𝑓 ( 𝑥 ) = 3 ( 𝑥 2 ) ( 𝑥 + 1 ) .

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

Q3:

Consider the power series 3 𝑥 .

Find the function 𝑓 represented by this series.

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

Determine the interval of convergence of the series.

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

Q4:

Consider the functions 𝑓 ( 𝑥 ) = ( 𝑥 1 ) 𝑛 ! and 𝑔 ( 𝑥 ) = ( 1 ) ( 𝑥 1 ) 𝑛 ! .

Find the power series of 1 2 [ 𝑓 ( 𝑥 ) + 𝑔 ( 𝑥 ) ] .

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

Find the power series of 1 2 [ 𝑓 ( 𝑥 ) 𝑔 ( 𝑥 ) ] .

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

Q5:

Let 𝑓 ( 𝑥 ) = 1 ( 𝑥 1 ) ( 𝑥 2 ) .

Construct a power series for the function 𝑓 ( 𝑥 ) .

  • A 1 1 2 𝑥
  • B 1 2 + 1 𝑥
  • C 1 2 1 𝑥
  • D 1 1 2 𝑥
  • E 1 + 1 2 𝑥

Find the interval of convergence of the power series.

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

Q6:

Use partial fractions to find the power series of the function 𝑓 ( 𝑥 ) = 3 ( 𝑥 + 1 ) ( 𝑥 + 4 ) .

  • A 1 1 4 𝑥
  • B ( 1 ) 1 1 4 𝑥
  • C 1 1 4 𝑥
  • D ( 1 ) 1 1 4 𝑥
  • E 1 1 4 𝑥

Q7:

Consider the functions 𝑓 ( 𝑥 ) = ( 2 𝑥 ) ( 2 𝑛 ) ! and 𝑔 ( 𝑥 ) = ( 2 𝑥 ) ( 2 𝑛 + 1 ) ! .

Find the power series of 𝑓 ( 𝑥 ) + 𝑔 ( 𝑥 ) .

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

Find the power series of 𝑓 ( 𝑥 ) 𝑔 ( 𝑥 ) .

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

Q8:

Multiply the series 1 1 + 𝑥 = ( 1 ) 𝑥 by itself to construct a series for 1 ( 1 + 𝑥 ) . Write the answer in sigma notation.

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

Q9:

Consider the power series 𝑓 ( 𝑥 ) = 1 4 𝑥 .

Find the function 𝑓 represented by this series.

  • A 𝑓 ( 𝑥 ) = 4 4 𝑥
  • B 𝑓 ( 𝑥 ) = 4 4 + 𝑥
  • C 𝑓 ( 𝑥 ) = 4 𝑥 4
  • D 𝑓 ( 𝑥 ) = 1 4 4 𝑥
  • E 𝑓 ( 𝑥 ) = 1 𝑥 + 4

Determine the interval of convergence of the series.

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

Let 𝑔 ( 𝑥 ) = 𝑓 ( 𝑥 ) . Find 𝑓 ( 𝑥 ) + 𝑔 ( 𝑥 ) in sigma notation.

  • A 2 1 4 𝑥
  • B 8 ( 1 ) 1 4 𝑥
  • C 1 2 𝑥
  • D ( 1 ) 1 4 𝑥
  • E 8 1 4 𝑥

Q10:

Find 𝑥 ( 2 𝑥 ) .

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

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