In Exercises 53–56, determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (n / (n² + 1))]

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Recognize that the given series is an alternating series of the form \(\sum_{n=1}^\infty (-1)^{n+1} a_n\) where \(a_n = \frac{n}{n^2 + 1}\). Since the terms alternate in sign, we can consider using the Alternating Series Estimation Theorem to bound the error.

Recall the Alternating Series Estimation Theorem 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., \(|S - S_N| \leq a_{N+1}\).

To ensure the error is less than 0.001, set up the inequality \(a_{N+1} = \frac{N+1}{(N+1)^2 + 1} < 0.001\) and solve for \(N\).

Rewrite the inequality as \(\frac{N+1}{(N+1)^2 + 1} < 0.001\) and multiply both sides by \(((N+1)^2 + 1)\) to get \((N+1) < 0.001 \times ((N+1)^2 + 1)\).

Solve this inequality for \(N\) to find the smallest integer \(N\) such that the error bound is satisfied. This \(N\) will be the number of terms needed to estimate the sum with an error less than 0.001.

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주요 개념

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Alternating Series

An alternating series is a series whose terms alternate in sign, typically involving (-1)^n or (-1)^(n+1). Such series often converge under specific conditions, and their behavior is key to estimating sums and errors.

Alternating Series Estimation Theorem

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. It allows us to determine how many terms are needed to achieve a desired accuracy.

Convergence and Term Behavior

For an alternating series to converge, the absolute value of its terms must decrease monotonically to zero. Understanding the behavior of the term n/(n² + 1) helps verify convergence and apply error bounds effectively.

In Exercises 53–56, determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001.∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (n / (n² + 1))]