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 , where .

Calculate .

  • A1
  • B
  • C0
  • D
  • E

Hence, determine whether the series converges or diverges.

  • AIt converges.
  • BIt diverges.

Q2:

Consider the series ∞ 𝑛 = 0 𝑛 ο„š π‘Ž , where π‘Ž = ( 𝑛 + 1 ) 6 𝑛 𝑛 2 𝑛 .

Calculate l i m 𝑛 β†’ ∞ 𝑛 | π‘Ž | . 1 𝑛

  • A 1 3 6
  • B 1 6
  • C0
  • D ∞
  • E6

Hence, determine whether the series converges or diverges.

  • AIt diverges.
  • BIt converges.

Q3:

A series ∞     ο„š π‘Ž satisfies l i m  β†’ ∞  | π‘Ž | = 1 .  

What can we conclude about the convergence of the series?

  • AThe series converges conditionally.
  • BThe series diverges.
  • CThe series converges absolutely.
  • DWe cannot conclude anything.

Q4:

Consider the series ∞ 𝑛 = 3 𝑛 ο„š ο€Ό 2 βˆ’ 𝑛 3 𝑛 + 1  .

Is this an alternating series?

  • Ayes
  • Bno

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

  • Aabsolutely convergent
  • Bdivergent
  • Cconditionally convergent

Q5:

Consider the series 1 + 1 2 + 1 2 + 1 2 + β‹― √  √  √ οŠͺ , where the term π‘Ž = 1 2  √  .

What is l i m  β†’ ∞     | π‘Ž | | π‘Ž | ?

What is l i m  β†’ ∞ √   ο„Ÿ 1 2 ?

Use L’Hopital’s rule to determine the value of the limit l i m l n  β†’ ∞ 𝐴 π‘₯ √ π‘₯ where 𝐴 > 0 is a constant.

What does the previous result tell you about the values of √ 𝑛 and l o g   𝑛 where 𝑛 β‰₯ 1 is an integer?

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

Is this series convergent or divergent?

  • Aconvergent
  • Bdivergent

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