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Worksheet: Ratio Test

Q1:

Consider the series ∞ 𝑛 = 0 𝑛 ο„š π‘Ž , where π‘Ž = 𝑏 ( 𝑛 + 𝑐 ) ! 𝑛 𝑛 for some integers 𝑏 , 𝑐 > 1 .

Calculate l i m 𝑛 β†’ ∞ 𝑛 + 1 𝑛 | | | π‘Ž π‘Ž | | | .

Hence, decide whether the series converges or diverges.

  • AIt converges.
  • BIt diverges.

Q2:

A series ∞ 𝑛 = 0 𝑛 ο„š π‘Ž satisfies l i m 𝑛 β†’ ∞ 𝑛 + 1 𝑛 | | | π‘Ž π‘Ž | | | = 1 . What can you conclude about the convergence of the series?

  • AThe series diverges.
  • BThe series converges conditionally.
  • CThe series converges absolutely.
  • DWe cannot conclude anything.

Q3:

Consider the series ∞ 𝑛 = 0 𝑛 𝑛 + 1 ο„š ( βˆ’ 2 ) 𝑛 3 .

Calculate l i m 𝑛 β†’ ∞ 𝑛 + 1 𝑛 | | | π‘Ž π‘Ž | | | .

  • A ∞
  • B0
  • C βˆ’ 2 3
  • D 2 3
  • E 3 2

Hence, determine whether the series converges or diverges.

  • AIt converges.
  • BIt diverges.

Q4:

True or False: The series ∞ 𝑛 = 1 2 ο„š 1 𝑛 + 1 is convergent by the ratio test.

  • Afalse
  • Btrue

Q5:

Consider the series ∞ 𝑛 = 0 𝑛 ο„š π‘Ž , where π‘Ž = ( 𝑛 + 𝑏 ) ! 𝑐 𝑛 𝑛 for some integers 𝑏 , 𝑐 > 1 .

Calculate l i m 𝑛 β†’ ∞ 𝑛 + 1 𝑛 | | | π‘Ž π‘Ž | | | .

  • A βˆ’ ∞
  • B0
  • C2
  • D ∞
  • E 1 2

Hence, decide whether the series converges or diverges.

  • AIt diverges.
  • BIt converges.