Worksheet: The Comparison Test for Series

In this worksheet, we will practice using the comparison test, where the convergence of a series is tested by comparing their terms to those of another whose convergence is known.

Q1:

Use the comparison test to decide whether the series 𝑛 = 1 𝑛 2 3 7 𝑛 + 1 . 1 is convergent or divergent.

  • Aconvergent
  • Bdivergent

Q2:

Let 𝑎 = 𝑛 + 1 𝑛 𝑛 + 4 𝑛 + 2 8 𝑛 2 3 2 . As 𝑛 , we see that 𝑎 1 𝑛 𝑛 , which suggests that 𝑛 = 1 𝑛 𝑎 is a divergent series. Verify this by finding the first integer 𝑛 where 𝑎 > 1 𝑛 𝑛 . You should check that this inequality remains true for all larger 𝑛 .

Q3:

Consider the series 𝑛 = 1 2 𝑛 𝑛 = 1 1 + 2 4 + 3 9 + s i n s i n s i n s i n .

Do the terms 𝑎 𝑛 of this sequence tend to 0 as 𝑛 ?

  • Ayes
  • Bno

Is the series convergent or divergent?

  • Aconvergent
  • Bdivergent

Q4:

Suppose 𝑛 = 1 𝑛 𝑎 is a series with the property that there is an integer 𝑁 such that 𝑛 𝑎 > 1 0 𝑛 8 for all indices 𝑛 𝑁 .

Does it follow that all the terms of the series are positive?

  • Ano
  • Byes

Does it follow that the number of negative terms is finite?

  • Ayes
  • Bno

Is the series convergent?

  • Ayes
  • BIt is not possible to tell.
  • Cno

Is the series convergent if the inequality is 𝑛 𝑎 < 1 0 2 𝑛 8 instead?

  • Ayes
  • Bno
  • CIt is not possible to tell.

Q5:

Use the comparison test to determine whether is convergent or divergent.

  • Adivergent
  • Bconvergent

Q6:

Use the comparison test to decide whether the series 𝑛 = 1 𝑛 𝑛 1 + 1 𝑛 𝑒 is convergent or divergent.

  • Aconvergent
  • Bdivergent

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