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(𝑛1)(𝑛1).

  • A2
  • BThe series diverges.
  • C 1 2
  • D1
  • E 𝜋 6

Q2:

All the terms of the series 1𝑛tan are positive when 𝑛 is large since tan𝑥>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 and lim𝑛𝑎=𝑐 with 𝑐>0. What does the limit comparison test tell us about the series 𝑎?

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

Q4:

Let 𝑎=𝑛7𝑛+7𝑛8 and consider the series 𝑎. We want to apply the limit comparison test.

Find 𝑝 so that lim𝑎 is a positive constant.

  • A2
  • B 1 2
  • C 2
  • D 1 2
  • E1

Is the series convergent or divergent?

  • Aconvergent
  • Bdivergent

Q5:

For the series 152, use the limit comparison test to determine whether the series converges or diverges.

  • AIt diverges.
  • BIt converges.

Q6:

For the series 1𝑒4, use the limit comparison test to determine whether the series converges or diverges.

  • AIt converges.
  • BIt diverges.

Q7:

For the series 1𝑛+1, use the limit comparison test to determine whether the series converges or diverges.

  • AIt diverges.
  • BIt converges.

Q8:

For the series 5+16, use the limit comparison test to determine whether the series converges or diverges.

  • AIt diverges.
  • BIt converges.

Q9:

For the series 𝑛2𝑛ln, use the limit comparison test to determine whether the series converges or diverges.

  • AIt converges.
  • BIt diverges.

Q10:

For the series 1+3𝑛ln, determine whether the series converges or diverges using the limit comparison test.

  • AIt diverges.
  • BIt converges.

Q11:

Using the limit comparison test, determine whether the series 14+6 is convergent or divergent.

  • AConvergent
  • BDivergent

Q12:

Use the limit comparison test to determine whether the series 2+52+4 is convergent or divergent.

  • ADivergent
  • BConvergent

Q13:

Use the limit comparison test to determine whether the series 2𝑛+6 is convergent or divergent.

  • AConvergent
  • BDivergent

Q14:

Use the limit comparison test to determine whether the series 12+3𝑛 is convergent or divergent.

  • AConvergent
  • BDivergent

Q15:

Use the limit comparison test to determine whether the series 𝑛+2𝑛𝑛+4𝑛 is convergent or divergent.

  • ADivergent
  • BConvergent

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