Question

For each of the following alternating series estimate the sum of the infinite series to within 10⁻⁶ . To do this you should first determine which partial sum will estimate the infinite sum to this accuracy, and then compute that partial sum.

(a) ∑[infinity]n=1 (−1)ⁿ⁺¹ 1/n⁶

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Answer 1

To estimate the sum of the given infinite alternating series within 10^-6, we find the smallest installment number n where 1/n^6 is less than our accuracy threshold. We then calculate the nth partial sum of the series as the estimate.

Step-by-step explanation:

The student is asking about the estimation of the sum of an infinite alternating series to a specified accuracy. Specifically, the series in question is ∑(− 1)^(n + 1) × (1/n^6), and we're looking to estimate the sum within 10^-6. To accomplish this, we use the fact that for an alternating series where the absolute value of the terms are decreasing, the error in using the nth partial sum to approximate the sum of the series is less than or equal to the absolute value of the first omitted term. Here, this means we need to find the smallest n such that 1/n^6 is less than 10^-6 to guarantee our required accuracy. Once we find such an n, we calculate the nth partial sum to estimate the infinite series.

To find the necessary n, we solve the inequality 1/n^6 < 10^-6 for n, which gives us n > 10. Thus, we'd calculate the sum from n=1 to n=10 to get our estimate. The partial sum is a finite calculation of alternating terms of the form (− 1)^(n + 1) × (1/n^6) up to n=10.