Worksheet: Remainder of an Alternating Series

In this worksheet, we will practice finding the error when approximating an alternating series by a finite term of the series.

Q1:

Calculate the partial sum 𝑆 for the least 𝑛 terms that guarantees that the sum of the first 𝑛 terms of the alternating series (1)5 differs from the infinite sum by 10 at most. Give your answer approximated to six decimal places.

Q2:

Consider the alternating series (1)𝑛.

Find the value of the first 𝑛 that guarantees that the sum of the first 𝑛 terms of the series differs from the infinite sum by 10 at most.

Calculate the partial sum 𝑆 for the 𝑛 terms in the previous part. Give your answer to five decimal places.

Q3:

Consider the alternating series (1)𝑒.

Find the value of 𝑛 that guarantees that the sum of the first 𝑛 terms of the series differs from the infinite sum by 10 at most.

Calculate the partial sum 𝑆 for the 𝑛 terms in the previous part. Give your answer to five decimal places.

Q4:

For the alternating series (1)𝑛, find the error bound when approximating the series by the first 20 terms. Give your answer to five decimal places.

Q5:

Calculate the partial sum 𝑆 for the least 𝑛 terms that guarantees that the sum of the first 𝑛 terms of the alternating series (1)2𝑛 differs from the infinite sum by 10 at most. Give your answer approximated to 3 decimal places.

Q6:

Is it possible to approximate the series 𝑆=(1)𝑛+13𝑛4 by summing its first 𝑛 terms? If yes, find the value of the first 𝑛 that guarantees that the sum of the first 𝑛 terms of the series 𝑆 differs from the infinite sum by at most 0.4.

  • AYes, 𝑛=12.
  • BYes, 𝑛=13.
  • CYes, 𝑛=14.
  • DYes, 𝑛=15.
  • ENo, the series diverges, so we cannot find an infinite sum to it.

Q7:

Which of the following series has a lower-bound error when approximated by the sum of its first 30 terms?

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

Q8:

Which of the following series requires summing the least number of terms so that the finite sum differs from the infinite sum by at most 10?

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

Q9:

Find the maximum error bound when approximating the series (1)3𝑛+7𝑛+1 by summing the first 20 terms. Round your answer to 5 decimal places.

Q10:

Find the lowest value of 𝑛 that guarantees that the sum of the first 𝑛 terms of the series 𝑆=1(𝑛)+1ln differs from the infinite sum by 0.26 at the most.

  • A𝑛=18
  • B𝑛=16
  • C𝑛=46
  • D𝑛=15
  • E𝑛=17

Q11:

Find the lowest value of 𝑛 that guarantees that the partial sum 𝑆 of the series 𝑆=(1)36+10 differs from the infinite sum by 10 at the most.

Q12:

The series (1)7𝑛 can be approximated by summing the first 7 terms. Find the maximum error bound of this approximation.

  • A7
  • B73
  • C724
  • D766
  • E722

Q13:

Find the lowest value of 𝑛 that guarantees that the partial sum 𝑆 of the series 𝑆=(1)(2𝑛)6𝑛ln differs from the infinite sum by at most 10.

Q14:

Which of the following series has a lower-bound error when approximated by the sum of its first 6 terms?

  • A(1)23
  • B(1)47
  • C(1)35
  • D(1)47
  • E(1)35

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