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Worksheet: Root Test

Q1:

A series 𝑎 satisfies l i m | 𝑎 | = 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.

Q2:

Consider the series 𝑛 = 0 𝑛 𝑎 , where 𝑎 = ( 𝑛 + 1 ) 6 𝑛 𝑛 2 𝑛 .

Calculate l i m 𝑛 𝑛 | 𝑎 | . 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:

Consider the series 𝑛 = 3 𝑛 2 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

Q4:

Consider the series 𝑛 = 0 𝑛 𝑎 , where 𝑎 = 𝑛 + 𝑛 𝑛 + 3 3 𝑛 + 6 𝑛 + 1 𝑛 3 2 3 2 2 𝑛 .

Calculate l i m 𝑛 𝑛 | 𝑎 | 1 𝑛 .

  • A1
  • B 1 3
  • C0
  • D 1 9
  • E

Hence, determine whether the series converges or diverges.

  • AIt converges.
  • BIt diverges.

Q5:

Consider the series 1 + 1 2 + 1 2 + 1 2 + 2 3 4 , where the term 𝑎 = 1 2 𝑛 𝑛 .

What is l i m 𝑛 𝑛 + 1 𝑛 | 𝑎 | | 𝑎 | ?

What is l i m 𝑛 𝑛 𝑛 1 2 ?

Use L’Hopital’s rule to determine the value of the limit l i m l n 𝑛 𝐴 𝑥 𝑥 where 𝐴 > 0 is a constant.

What does the previous result tell you about the values of 𝑛 and 𝑛 l o g 2 2 where 𝑛 1 is an integer?

  • AIt tells us that 𝑛 > ( 𝑛 ) l o g 2 2 for all large values of 𝑛 .
  • BIt tells us nothing.
  • CIt tells us that 𝑛 and ( 𝑛 ) l o g 2 2 are both zero if 𝑛 is large enough.
  • DIt tells us that 𝑛 > ( 𝑛 ) l o g 2 2 for all values of 𝑛 .
  • EIt tells us that 𝑛 < ( 𝑛 ) l o g 2 2 for all large values of 𝑛 .

Is this series convergent or divergent?

  • Aconvergent
  • Bdivergent