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+1𝑛ln.

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

  • Aln𝑀𝑀+1
  • Bln𝑀𝑀1
  • Cln𝑀
  • Dln(𝑀+1)
  • Eln(𝑀1)

Is the series convergent?

  • Ano
  • Byes

Q2:

Find the partial sum for the series 𝑒𝑒.

  • A𝑆=𝑒𝑒
  • B𝑆=𝑒𝑒
  • C𝑆=𝑒
  • D𝑆=𝑒+𝑒
  • E𝑆=𝑒

Is the series convergent or divergent?

  • AConvergent
  • BDivergent

Q3:

Find the partial sum for the series 212.

  • A𝑆=14
  • B𝑆=112
  • C𝑆=4
  • D𝑆=4112
  • E𝑆=2112

Is the series convergent or divergent?

  • AConvergent
  • BDivergent

Q4:

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

  • A𝑆=𝑛+72(𝑛+3)
  • B𝑆=𝑛2(𝑛+4)
  • C𝑆=𝑛2(𝑛+3)
  • D𝑆=2𝑛+12
  • E𝑆=𝑛+82(𝑛+4)

Is the series convergent or divergent?

  • AConvergent
  • BDivergent

Q5:

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

  • A𝑆=3(21)
  • B𝑆=6
  • C𝑆=3(21)
  • D𝑆=3(2)
  • E𝑆=16

Is the series convergent or divergent?

  • AConvergent
  • BDivergent

Q6:

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

  • AIt converges.
  • BIt diverges.

Q7:

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

  • AIt diverges.
  • BIt converges.

Q8:

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

  • A𝑘𝑘+1
  • B2𝑘𝑘+1
  • C𝑘2𝑘+2
  • D2𝑘𝑘1
  • E𝑘+2𝑘+1

Q9:

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

  • A𝑆=4𝑛(𝑛+2)
  • B𝑆=4𝑛(𝑛+2)
  • C𝑆=4𝑛
  • D𝑆=4𝑛
  • E𝑆=4(𝑛+2)

Does the series converge or diverge?

  • AThe series converges.
  • BThe series diverges.

Q10:

Find the partial sum of the series 𝜋𝑛+𝜋𝑛+1+𝜋2sincos.

  • A𝑆=𝜋𝑛+1sin
  • B𝑆=1𝜋𝑛sin
  • C𝑆=1+𝜋𝑛sin
  • D𝑆=𝜋𝑛+1sin
  • E𝑆=1𝜋𝑛+1sin

Is the series convergent or divergent?

  • AConvergent
  • BDivergent

Q11:

Find the partial sum of the series 5𝑛𝑛+5𝑛+6𝑛.

  • A𝑆=535𝑛+3
  • B𝑆=5𝑛35𝑛+3
  • C𝑆=5𝑛+35𝑛+2
  • D𝑆=5𝑛+353
  • E𝑆=5𝑛+25𝑛+3

Does the series converge or diverge?

  • AThe series converges.
  • BThe series diverges.

Q12:

Find the partial sum of the series 815.

  • A𝑆=50115
  • B𝑆=50(1(5))
  • C𝑆=8115
  • D𝑆=8115
  • E𝑆=50115

Does the series converge or diverge?

  • AThe series diverges.
  • BThe series converges.

Q13:

Find the partial sum of the series 1(2𝑛+1)(2𝑛1).

  • A𝑆=22𝑛+1
  • B𝑆=𝑛2𝑛+1
  • C𝑆=𝑛2(2𝑛+1)
  • D𝑆=2𝑛+1𝑛
  • E𝑆=2𝑛+12𝑛

Is the series convergent or divergent?

  • AConvergent
  • BDivergent

Q14:

Find the partial sum of the series (𝑛+1)(𝑛)tantan.

  • A𝑆=(𝑛+1)(1)tantan
  • B𝑆=(𝑛+1)(𝑛)tantan
  • C𝑆=(𝑛)+(1)tantan
  • D𝑆=(𝑛)(𝑛+1)tantan
  • E𝑆=(𝑛)(1)tantan

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