Approximate Alternating Series Sum

Approximate the sum of the series $(-1)^{n+1}/n^3$ correct to three decimal places.

This problem explores the concept of estimating the sum of an alternating series to a specified accuracy. Alternating series are characterized by their terms changing sign and typically converging more rapidly than non-alternating series. The Alternating Series Test is a critical tool when analyzing these kinds of series. A series converges if its terms decrease in magnitude and tend to zero. Once convergence is established, the Alternating Series Estimation Theorem can be applied to approximate the sum to a desired precision. This theorem states that the error in approximating the sum of an alternating series by its first n terms is less than or equal to the absolute value of the (n+1)th term.

In practice, approximating the sum involves summing the terms in sequence until the next term in the series is smaller than the desired level of accuracy. It's important to recognize why alternating series that meet the test's conditions have smaller errors earlier in their sequence. This is because each added term reduces the error in alternating signs. Understanding these concepts aids in solving approximation problems and reinforces the application of convergence tests alongside error estimation strategies. Such skills are crucial in mathematical analysis where precise calculations are necessary.