Worksheet: Partial Sums

In this worksheet, we will practice finding the nth partial sum of a series, and determining the convergence or divergence of the series from the limit of its partial sum.

Q1:

Consider the series 𝑛 + 1 𝑛 l n .

Give an exact expression for the partial sum 𝑛 + 1 𝑛 l n .

  • A l n 𝑀 𝑀 + 1
  • B l n 𝑀
  • C l n ( 𝑀 1 )
  • D l n ( 𝑀 + 1 )
  • E l n 𝑀 𝑀 1

Is the series convergent?

  • Ayes
  • Bno

Q2:

Find the partial sum for the series 𝑒 𝑒 .

  • A 𝑆 = 𝑒
  • B 𝑆 = 𝑒
  • C 𝑆 = 𝑒 𝑒
  • D 𝑆 = 𝑒 𝑒
  • E 𝑆 = 𝑒 + 𝑒

Is the series convergent or divergent?

  • AConvergent
  • BDivergent

Q3:

Find the partial sum for the series 2 1 2 .

  • A 𝑆 = 4
  • B 𝑆 = 4 1 1 2
  • C 𝑆 = 2 1 1 2
  • D 𝑆 = 1 1 2
  • E 𝑆 = 1 4

Is the series convergent or divergent?

  • AConvergent
  • BDivergent

Q4:

Find the partial sum for the series 2 ( 𝑛 + 3 ) ( 𝑛 + 4 ) .

  • A 𝑆 = 𝑛 + 8 2 ( 𝑛 + 4 )
  • B 𝑆 = 2 𝑛 + 1 2
  • C 𝑆 = 𝑛 2 ( 𝑛 + 4 )
  • D 𝑆 = 𝑛 2 ( 𝑛 + 3 )
  • E 𝑆 = 𝑛 + 7 2 ( 𝑛 + 3 )

Is the series convergent or divergent?

  • AConvergent
  • BDivergent

Q5:

Find the partial sum for the series 3 ( 2 ) .

  • A 𝑆 = 1 6
  • B 𝑆 = 3 ( 2 1 )
  • C 𝑆 = 3 ( 2 1 )
  • D 𝑆 = 3 ( 2 )
  • E 𝑆 = 6

Is the series convergent or divergent?

  • AConvergent
  • BDivergent

Q6:

Use the sequence of partial sums to determine whether the series 𝑛 𝑛 + 1 converges or diverges.

  • AIt converges.
  • BIt diverges.

Q7:

Use the sequence of partial sums to determine whether the series 1 𝑛 + 1 converges or diverges.

  • AIt diverges.
  • BIt converges.

Q8:

Write the expression 2 + 1 + 2 3 + 1 2 + 2 5 + as an infinite series using sigma notation.

  • A 2 𝑛 !
  • B 𝑛 2
  • C 2 𝑛 + 1
  • D 1 𝑛 !
  • E 2 𝑛

Q9:

Write the expression 1 1 2 + 1 3 1 4 + as an infinite series, using sigma notation.

  • A 1 𝑛
  • B ( 1 ) 𝑛
  • C 1 𝑛
  • D ( 1 ) 𝑛
  • E ( 1 ) 𝑛

Q10:

Evaluate the sequence of partial sums 𝑆 for the series 2 𝑛 ( 𝑛 + 1 ) .

  • A 𝑘 + 2 𝑘 + 1
  • B 2 𝑘 𝑘 1
  • C 𝑘 2 𝑘 + 2
  • D 𝑘 𝑘 + 1
  • E 2 𝑘 𝑘 + 1

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