Worksheet: The nth Term Divergence Test

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In this worksheet, we will practice applying the nth term test for divergence which states that a series diverges if the limit of its nth term does not approach zero.

Q1:

True or False: If 𝑎 → 0 as 𝑛 → ∞ , then ∞ 𝑎 is convergent.

  • Afalse
  • Btrue

Q2:

What can we conclude by applying the 𝑛 t h term divergence test in the series ∞ ( 5 𝑛 + 3 ) ( 7 𝑛 + 2 ) ?

  • AThe series diverges.
  • BThe divergence test is inconclusive.

Q3:

What can we conclude by applying the 𝑛 t h term divergence test in the series ∞ 3 𝑛 √ 6 𝑛 + 4 𝑛 + 5 ?

  • AThe series diverges.
  • BThe divergence test is inconclusive.

Q4:

What can we conclude by applying the 𝑛 t h term divergence test in the series ∞ 3 𝑛 c o s ?

  • AThe series diverges.
  • BThe divergence test is inconclusive.

Q5:

What can we conclude by applying the 𝑛 t h term divergence test in the series ∞ 2 𝑛 3 𝑛 l n ?

  • AThe divergence test is inconclusive.
  • BThe series converges.

Q6:

What can we conclude by applying the 𝑛 t h term divergence test in the series ∞ 2 𝑛 6 𝑛 + 4 ?

  • AThe divergence test is inconclusive.
  • BThe series diverges.

Q7:

True or False: If 𝑎 → 0 as 𝑛 → ∞ , then the 𝑛 t h term divergence test fails for ∞ 𝑎 .

  • ATrue
  • BFalse

Q8:

Using the 𝑛 t h term test, determine whether the series ∞ 𝑛 𝑛 + 1 is divergent or the test fails.

  • AThe test fails.
  • BThe series is divergent.

Q9:

Using the nth term test, determine whether the series ∞ 4 3 is divergent or the test fails.

  • AThe series is divergent.
  • BThe test fails.

Q10:

Using the 𝑛 t h term test, determine whether the series ∞ ( 𝑛 + 2 ) ! ( 2 𝑛 ) ! is divergent or the test fails.

  • AThe series is divergent.
  • BThe test fails.

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