Worksheet: Root Test

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In this worksheet, we will practice determining if a series is convergent or divergent using the root test.

Q1:

Consider the series ∞𝑎, where 𝑎=𝑛+𝑛−𝑛+33𝑛+6𝑛+1.

Calculate lim→∞|𝑎|.

  • A19
  • B0
  • C1
  • D∞
  • E13

Hence, determine whether the series converges or diverges.

  • AIt converges.
  • BIt diverges.

Q2:

Consider the series ∞𝑎, where 𝑎=(𝑛+1)6.

Calculate lim→∞|𝑎|.

  • A16
  • B6
  • C0
  • D136
  • E∞

Hence, determine whether the series converges or diverges.

  • AIt converges.
  • BIt diverges.

Q3:

A series ∞𝑎 satisfies lim→∞|𝑎|=1.

What can we conclude about the convergence of the series?

  • AWe cannot conclude anything.
  • BThe series diverges.
  • CThe series converges absolutely.
  • DThe series converges conditionally.

Q4:

Consider the series ∞2−𝑛3𝑛+1.

Is this an alternating series?

  • Ayes
  • Bno

Is this series absolutely convergent, conditionally convergent, or divergent?

  • Aabsolutely convergent
  • Bconditionally convergent
  • Cdivergent

Q5:

Consider the series 1+12+12+12+⋯√√√, where the term 𝑎=12√.

What is lim→∞|𝑎||𝑎|?

What is lim→∞√12?

Use l’Hopital’s rule to determine the value of the limit limln→∞𝐴𝑛√𝑛, where 𝐴>0 is a constant.

What does the previous result tell you about the values of √𝑛 and log𝑛, where 𝑛≥1 is an integer?

  • AIt tells us nothing.
  • BIt tells us that √𝑛>𝑛log for all values of 𝑛.
  • CIt tells us that √𝑛<𝑛log for all large values of 𝑛.
  • DIt tells us that √𝑛>𝑛log for all large values of 𝑛.
  • EIt tells us that √𝑛 and log𝑛 are both zero if 𝑛 is large enough.

Using the comparison test, is this series convergent or divergent?

  • AConvergent
  • BDivergent

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