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 lim|||𝑎𝑎|||.

  • A
  • B12
  • C
  • D2
  • E0

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 lim|||𝑎𝑎|||.

Hence, decide whether the series converges or diverges.

  • AIt diverges.
  • BIt converges.

Q3:

Consider the series (2)𝑛3.

Calculate lim|||𝑎𝑎|||.

  • A
  • B32
  • C23
  • D23
  • E0

Hence, determine whether the series converges or diverges.

  • AIt converges.
  • BIt diverges.

Q4:

A series 𝑎 satisfies lim|||𝑎𝑎|||=1. What can you conclude about the convergence of the series?

  • AThe series converges conditionally.
  • BThe series diverges.
  • CWe cannot conclude anything.
  • DThe series converges absolutely.

Q5:

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

  • Afalse
  • Btrue

Q6:

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

  • AIt converges.
  • BIt diverges.

Q7:

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

  • AIt converges.
  • BIt diverges.

Q8:

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

  • AIt diverges.
  • BIt converges.

Q9:

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

  • AIt diverges.
  • BIt converges.

Q10:

Consider the series 𝑛3𝑛!. Determine whether the series converges or diverges.

  • AIt converges.
  • BIt diverges.

Q11:

Using the ratio test, determine whether the series (2)𝑛𝑛! is convergent, divergent, or whether the test is inconclusive.

  • AThe series is convergent.
  • BThe test is inconclusive.
  • CThe series is divergent.

Q12:

Given that 𝑘 is a positive integer, find the set of values of 𝑘 for which the series 2(𝑛!)(𝑘𝑛)! is convergent.

  • A𝑘>2
  • B𝑘=2
  • C𝑘2
  • D𝑘2
  • E𝑘<2

Q13:

Using the ratio test, determine whether the series 3(2𝑛)! is convergent, divergent, or whether the test is inconclusive.

  • AThe series is convergent.
  • BThe test is inconclusive.
  • CThe series is divergent.

Q14:

Using the ratio test, determine whether the series 2𝑛𝑛! is convergent, divergent, or whether the test is inconclusive.

  • AThe test is inconclusive.
  • BThe series is divergent.
  • CThe series is convergent.

Q15:

The terms of a series are defined by the equations 𝑎=7 and 𝑎=3𝑛+15(2𝑛5). Using the ratio test, determine whether the series 𝑎 is convergent, divergent, or whether the test is inconclusive.

  • AThe series is divergent.
  • BThe test is inconclusive.
  • CThe series is convergent.

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