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Lesson: Limit Comparison Test

In this lesson, we will learn how to determine whether any series whose nth term is a rational function is convergent by comparing its terms to those of another whose convergence is known using limits.

Worksheet • 4 Questions

Q1:

If possible, find the sum of the series ∞ 𝑛 = 1 3 ο„š 𝑛 ( 𝑛 βˆ’ 1 ) ( 𝑛 βˆ’ 1 ) .

  • AThe series diverges.
  • B 1 2
  • C πœ‹ 6 2
  • D2
  • E1

Q2:

All the terms of the series ∞ 𝑛 = 1 1 . 5 ο„š ο€Ό 1 𝑛  t a n are positive when 𝑛 is large since t a n π‘₯ > 0 when π‘₯ > 0 and close to 0. Use the limit comparison test to decide whether the series is convergent or divergent.

  • Adivergent
  • Bconvergent

Q3:

Suppose π‘Ž > 0 𝑛 and l i m 𝑛 β†’ ∞ 𝑛 √ 𝑛 π‘Ž = 𝑐 with 𝑐 > 0 . What does the limit comparison test tell us about the series ∞ 𝑛 = 1 𝑛 ο„š π‘Ž ?

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

Q4:

Let π‘Ž = 𝑛 βˆ’ 7 √ 𝑛 + 7 𝑛 βˆ’ 8 𝑛 3 and consider the series ∞ 𝑛 = 1 𝑛 ο„š π‘Ž . We want to apply the limit comparison test.

Find 𝑝 so that l i m 𝑛 β†’ ∞ 𝑛 1 𝑛 π‘Ž 𝑝 is a positive constant.

  • A 1 2
  • B βˆ’ 2
  • C2
  • D βˆ’ 1 2
  • E1

Is the series convergent or divergent?

  • Aconvergent
  • Bdivergent
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