Worksheet: Comparison Test for Series

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 comparison test.

Q1:

Use the comparison test to decide whether the series 237𝑛+1.1 is convergent or divergent.

  • Aconvergent
  • Bdivergent

Q2:

Let 𝑎=𝑛+1𝑛𝑛+4𝑛+28. As 𝑛, we see that 𝑎1𝑛, which suggests that 𝑎 is a divergent series. Verify this by finding the first integer 𝑛 where 𝑎>1𝑛. You should check that this inequality remains true for all larger 𝑛.

Q3:

Consider the series 𝑛𝑛=11+24+39+sinsinsinsin.

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

  • Ano
  • Byes

Is the series convergent or divergent?

  • Adivergent
  • Bconvergent

Q4:

Suppose 𝑎 is a series with the property that there is an integer 𝑁 such that 𝑛𝑎>10 for all indices 𝑛𝑁.

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?

  • Ano
  • BIt is not possible to tell.
  • Cyes

Is the series convergent if the inequality is 𝑛𝑎<10 instead?

  • Ano
  • Byes
  • CIt is not possible to tell.

Q5:

Use the comparison test to determine whether 1𝑛ln is convergent or divergent.

  • Aconvergent
  • Bdivergent

Q6:

Use the comparison test to decide whether the series 1+1𝑛𝑒 is convergent or divergent.

  • Adivergent
  • Bconvergent

Q7:

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

  • ADivergent
  • BConvergent

Q8:

Use the comparison test to determine whether the series 3𝑛 is convergent or divergent.

  • ADivergent
  • BConvergent

Q9:

Use the comparison test to determine whether the series 𝑛+1𝑛+3 is convergent or divergent.

  • AConvergent
  • BDivergent

Q10:

Use the comparison test to determine whether the series 1𝑛+2 is convergent or divergent.

  • ADivergent
  • BConvergent

Q11:

Use the comparison test to determine whether the series 2𝑛3𝑛+7𝑛 is convergent or divergent.

  • ADivergent
  • BConvergent

Q12:

Use the comparison test to determine whether the series 13 is convergent or divergent.

  • ADivergent
  • BConvergent

Q13:

Use the comparison test to determine whether the series 𝑒𝑛+(𝑛)sin is convergent or divergent.

  • AConvergent
  • BDivergent

Q14:

Use the comparison test to determine whether the series 13+5 is convergent or divergent.

  • ADivergent
  • BConvergent

Q15:

Use the comparison test to determine whether the series 43𝑛+𝑛+5𝑛+2 is convergent or divergent.

  • AConvergent
  • BDivergent

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