Worksheet: Limit Comparison Test

In this worksheet, we will practice determining whether a series is convergent or divergent by comparing it to a series of known convergence using the limit comparison test.

Q1:

If possible, find the sum of the series 𝑛 = 1 3 𝑛 ( 𝑛 1 ) ( 𝑛 1 ) .

  • A1
  • B 𝜋 6 2
  • C2
  • DThe series diverges.
  • E 1 2

Q2:

All the terms of the series 𝑛 = 1 1 . 5 1 𝑛 t a n are positive when 𝑛 is large since t a n 𝑥 > 0 when 𝑥 > 0 and close to 0. Use the limit comparison test to decide whether the series is convergent or divergent.

  • Aconvergent
  • Bdivergent

Q3:

Suppose 𝑎 > 0 𝑛 and l i m 𝑛 𝑛 𝑛 𝑎 = 𝑐 with 𝑐 > 0 . What does the limit comparison test tell us about the series 𝑛 = 1 𝑛 𝑎 ?

  • AIt tells us that it is alternating.
  • BIt tells us that it is convergent.
  • CIt tells us that it is constant.
  • DIt tells us that it is divergent.
  • EIt tells us nothing.

Q4:

Let 𝑎 = 𝑛 7 𝑛 + 7 𝑛 8 𝑛 3 and consider the series 𝑛 = 1 𝑛 𝑎 . We want to apply the limit comparison test.

Find 𝑝 so that l i m 𝑛 𝑛 1 𝑛 𝑎 𝑝 is a positive constant.

  • A1
  • B2
  • C 1 2
  • D 1 2
  • E 2

Is the series convergent or divergent?

  • Adivergent
  • Bconvergent

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