39–44. {Use of Tech} Estimating infinite series Estimate the value of the following convergent series with an absolute error less than 10⁻³.


∑ (k = 1 to ∞) (−1)ᵏ / k⁵

Verified step by step guidance

1. Recognize that the series given is an alternating series of the form \(\sum_{k=1}^\infty \frac{(-1)^k}{k^5}\), where the terms decrease in absolute value and approach zero as \(k\) increases, which means the series converges by the Alternating Series Test.
2. To estimate the sum with an absolute error less than \$10^{-3}\(, use the Alternating Series Estimation Theorem, which states that the absolute error when approximating the sum by the first \)n\( terms is less than or equal to the absolute value of the first omitted term, i.e., \)|R_n| \leq |a_{n+1}|$.
3. Find the smallest integer \(n\) such that the absolute value of the \((n+1)\)-th term satisfies \(\left| \frac{(-1)^{n+1}}{(n+1)^5} \right| = \frac{1}{(n+1)^5} < 10^{-3}\).
4. Once \(n\) is found, compute the partial sum \(S_n = \sum_{k=1}^n \frac{(-1)^k}{k^5}\) by adding the first \(n\) terms of the series.
5. This partial sum \(S_n\) will be an estimate of the infinite series with an absolute error less than \$10^{-3}$, as guaranteed by the Alternating Series Estimation Theorem.

Key Concepts

Convergence of Infinite Series

An infinite series converges if the sum of its terms approaches a finite limit as the number of terms increases. For alternating series like this one, the Alternating Series Test helps determine convergence by checking if terms decrease in absolute value and approach zero.

Alternating Series Estimation Theorem

This theorem states that the absolute error when approximating an alternating series by its partial sum is less than or equal to the absolute value of the first omitted term. It allows us to estimate how many terms are needed to achieve a desired error bound.

Absolute Error and Partial Sums

Absolute error measures the difference between the true sum and the partial sum approximation. By calculating partial sums and comparing the size of the next term, we can ensure the error is below a specified threshold, such as 10⁻³ in this problem.