Worksheet: Ratio Test

In this worksheet, we will practice determining if a series is convergent or divergent using the ratio test.

Q1:

Consider the series 𝑎 , where 𝑎 = ( 𝑛 + 𝑏 ) ! 𝑐 for some integers 𝑏 , 𝑐 > 1 .

Calculate l i m | | | 𝑎 𝑎 | | | .

  • A2
  • B
  • C
  • D0
  • E 1 2

Hence, decide whether the series converges or diverges.

  • AIt diverges.
  • BIt converges.

Q2:

Consider the series 𝑎 , where 𝑎 = 𝑏 ( 𝑛 + 𝑐 ) ! for some integers 𝑏 , 𝑐 > 1 .

Calculate l i m | | | 𝑎 𝑎 | | | .

Hence, decide whether the series converges or diverges.

  • AIt diverges.
  • BIt converges.

Q3:

Consider the series ( 2 ) 𝑛 3 .

Calculate l i m | | | 𝑎 𝑎 | | | .

  • A 2 3
  • B0
  • C 2 3
  • D
  • E 3 2

Hence, determine whether the series converges or diverges.

  • AIt converges.
  • BIt diverges.

Q4:

A series 𝑎 satisfies l i m | | | 𝑎 𝑎 | | | = 1 . What can you conclude about the convergence of the series?

  • AThe series converges conditionally.
  • BWe cannot conclude anything.
  • CThe series diverges.
  • DThe series converges absolutely.

Q5:

True or False: The series 1 𝑛 + 1 is convergent by the ratio test.

  • Atrue
  • Bfalse

Q6:

Consider the series ( 1 ) ( 𝑛 ! ) ( 3 𝑛 ) ! . Determine whether the series converges or diverges.

  • AIt diverges.
  • BIt converges.

Q7:

Consider the series 3 𝑛 ! . Determine whether the series converges or diverges.

  • AIt converges.
  • BIt diverges.

Q8:

Consider the series ( 4 𝑛 ) ! 𝑛 . Determine whether the series converges or diverges.

  • AIt converges.
  • BIt diverges.

Q9:

Consider the series 𝑛 𝑛 ! ( 1 ) ( 𝑛 ! ) ( 3 𝑛 ) ! . Determine whether the series converges or diverges.

  • AIt diverges.
  • BIt converges.

Q10:

Consider the series 𝑛 3 𝑛 ! . Determine whether the series converges or diverges.

  • AIt diverges.
  • BIt converges.

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