{Use of Tech} Error in a finite alternating sum
How many terms of the series ∑ (from k = 1 to ∞)(−1)ᵏ⁺¹ / k⁴ must be summed to ensure that the approximation error is less than 10⁻⁸?

Verified step by step guidance

1. Recognize that the given series is an alternating series of the form \( \sum_{k=1}^\infty (-1)^{k+1} \frac{1}{k^4} \), which satisfies the conditions of the Alternating Series Test: the terms \( \frac{1}{k^4} \) are positive, decreasing, and approach zero as \( k \to \infty \).
2. Recall 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., \( |R_n| \leq \left| a_{n+1} \right| \).
3. Set up the inequality for the error bound: \( \left| a_{n+1} \right| = \frac{1}{(n+1)^4} < 10^{-8} \), since the error must be less than \( 10^{-8} \).
4. Solve the inequality \( \frac{1}{(n+1)^4} < 10^{-8} \) by taking reciprocals and then the fourth root: \( (n+1)^4 > 10^{8} \) which implies \( n+1 > (10^{8})^{1/4} \).
5. Calculate \( (10^{8})^{1/4} = 10^{8/4} = 10^{2} = 100 \), so \( n+1 > 100 \) and therefore \( n \geq 100 \). This means you must sum at least 100 terms to ensure the error is less than \( 10^{-8} \).

Key Concepts

Alternating Series and Alternating Series Test

An alternating series is a series whose terms alternate in sign. The Alternating Series Test states that if the absolute value of the terms decreases monotonically to zero, the series converges. This test also provides a way to estimate the error when approximating the sum by partial sums.

Error Bound for Alternating Series

For an alternating series that meets the conditions of the Alternating Series Test, the error made by stopping at the nth term is less than or equal to the absolute value of the (n+1)th term. This allows us to control the approximation error by choosing a sufficient number of terms.

Convergence Rate of p-Series with p=4

The given series involves terms of the form 1/k⁴, which is a p-series with p=4. Since p > 1, the series converges rapidly. Understanding the rate of decay of terms helps in estimating how many terms are needed to achieve a desired accuracy.