8–32. {Use of Tech} Estimating errors in partial sums For each of the following convergent alternating series, evaluate the nth partial sum for the given value of n. Then use Theorem 10.18 to find an upper bound for the error |S − Sₙ| in using the nth partial sum Sₙ to estimate the value of the series S.


∑ (k = 1 to ∞) (−1)ᵏ / k⁴; n = 4

Verified step by step guidance

1. Identify the given alternating series: \( \sum_{k=1}^{\infty} \frac{(-1)^k}{k^4} \). This is an alternating series where the terms are \( a_k = \frac{1}{k^4} \).
2. Calculate the 4th partial sum \( S_4 \) by summing the first 4 terms of the series: \( S_4 = \sum_{k=1}^4 \frac{(-1)^k}{k^4} = \frac{(-1)^1}{1^4} + \frac{(-1)^2}{2^4} + \frac{(-1)^3}{3^4} + \frac{(-1)^4}{4^4} \).
3. Recall Theorem 10.18 (Alternating Series Estimation Theorem), which states that the absolute error \( |S - S_n| \) when approximating the sum \( S \) by the partial sum \( S_n \) is less than or equal to the absolute value of the first omitted term: \( |S - S_n| \leq |a_{n+1}| \).
4. Find the upper bound for the error by evaluating the absolute value of the next term after the 4th partial sum: \( |a_5| = \left| \frac{(-1)^5}{5^4} \right| = \frac{1}{5^4} \).
5. Summarize the results: the 4th partial sum \( S_4 \) approximates the series sum \( S \), and the error in this approximation is at most \( \frac{1}{5^4} \).

Key Concepts

Alternating Series

An alternating series is a series whose terms alternate in sign, typically expressed as (-1)^k times a positive term. Such series often converge under specific conditions, especially when the absolute value of terms decreases monotonically to zero. Understanding the structure of alternating series is crucial for applying convergence tests and error estimates.

Partial Sums of a Series

A partial sum Sₙ is the sum of the first n terms of a series and serves as an approximation to the series' total sum S. Calculating partial sums helps estimate the value of infinite series by truncating after a finite number of terms, which is essential for practical computations and error analysis.

Error Bound for Alternating Series (Theorem 10.18)

Theorem 10.18 states that for a convergent alternating series with decreasing terms, the absolute error |S − Sₙ| in approximating the sum by the nth partial sum is at most the absolute value of the (n+1)th term. This provides a straightforward way to estimate the maximum error when using partial sums.