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 1 0 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 1 0 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 1 0 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 ) 2 𝑛 differs from the infinite sum by 1 0 at most. Give your answer approximated to 3 decimal places.

Q6:

Is it possible to approximate the series 𝑆 = ( 1 ) 𝑛 + 1 3 𝑛 4 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 .

  • ANo, the series diverges, so we cannot find an infinite sum to it.
  • BYes, 𝑛 = 1 3 .
  • CYes, 𝑛 = 1 4 .
  • DYes, 𝑛 = 1 2 .
  • EYes, 𝑛 = 1 5 .

Q7:

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

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

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 1 0 ?

  • 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 ) 3 𝑛 + 7 𝑛 + 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 ) + 1 l n differs from the infinite sum by 0.26 at the most.

  • A 𝑛 = 1 7
  • B 𝑛 = 1 5
  • C 𝑛 = 1 6
  • D 𝑛 = 1 8
  • E 𝑛 = 4 6

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