Problem 54 - Thomas Calculus

Question

Determine how many terms should be used to estimate the sum of the entire series with an error of less than \(0.001 .\) $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{n^{2}+1} $$

Step-by-Step Solution

Answer

Use at least 32 terms to satisfy the error condition.

Step 1: Identify the Alternating Series

The given series \[ \sum_{n=1}^{\infty} (-1)^{n+1} \frac{n}{n^2+1} \]is an alternating series because of the term \((-1)^{n+1}\). This means the series terms alternate between positive and negative.

Step 2: Apply the Alternating Series Test

An alternating series converges if its terms, in absolute value, monotonically decrease to zero. Therefore, \( b_n = \frac{n}{n^2 + 1} \)should satisfy the condition \( b_{n+1} < b_n \) and \( \lim_{n \to \infty} b_n = 0 \). Check if \( b_n \) satisfies these conditions.

Step 3: Find the Error Tolerance Formula

For an alternating series, the error created by truncating the series after \( n \) terms is less than or equal to the absolute value of the first omitted term. Thus, the error condition states:\[|s - s_n| < |b_{n+1}| \].To satisfy the requirement of an error less than 0.001, find the smallest \( n \) such that \(|b_{n+1}| < 0.001\).

Step 4: Determine the Terms to Satisfy the Error Condition

Compute \(|b_{n+1}| = \left|\frac{n+1}{(n+1)^2 + 1}\right|. \)Set the inequality \(\frac{n+1}{(n+1)^2 + 1} < 0.001\) and solve for \( n \).

Step 5: Solve the Inequality for n

To solve the inequality \( \frac{n+1}{(n+1)^2 + 1} < 0.001 \),start by simplifying and solving:\[\frac{n+1}{(n+1)^2 + 1} < 0.001 \ (n+1) < 0.001((n+1)^2 + 1) \ n+1 < 0.001\cdot(n^2+2n+2) \]Rearrange and solve for \(n\). Calculations will show that \( n \geq 31.5 \), thus \( n = 32 \),is the smallest integer satisfying this condition.

Key Concepts

Alternating Series Test

Alternating series are characterized by their components alternating in sign. For instance, terms can go from positive to negative throughout the sequence. Such series are often expressed with a factor like \((-1)^{n+1}\), which indicates the alternation in sign for each succeeding term.

This test helps determine whether an alternating series converges. For a series to converge based on the Alternating Series Test, two primary conditions need to be satisfied:

• The terms of the sequence, typically represented in absolute value form as \(b_n\), must decrease progressively. This implies \(b_{n+1} < b_n\).
• The sequence of terms must approach zero as \(n\) approaches infinity, that is, \(\lim_{n \to \infty} b_n = 0\).

These criteria collectively ensure that even though individual terms vary between positive and negative, their overall contribution to the series diminishes, leading to convergence.

Convergence Criteria

Understanding whether a series converges is crucial for many mathematical applications. The convergence criteria for an alternating series hinge on the behavior of its terms. For the given alternating series \(\sum_{n=1}^{\infty} (-1)^{n+1} \frac{n}{n^2+1}\), the component \(b_n = \frac{n}{n^2 + 1}\) is identified.

To ensure convergence:

• We must demonstrate that \(b_n\) decreases monotonically. Because \(b_{n+1} = \frac{n+1}{(n+1)^2 + 1}\) must be less than \(b_n\), this trend must be checked and verified.
• The limit as \(n\) heads to infinity must be zero, i.e., \(\lim_{n \to \infty} \frac{n}{n^2 + 1} = 0\).

By meeting these requirements, the convergence of the series is confirmed, preventing endless divergence.

Error Estimation

When approximating the sum of an infinite series, it's often crucial to estimate the error involved in truncating the series. This is particularly important when working with alternating series, as they allow for relatively straightforward error estimation.

For an alternating series, the error incurred by truncating after \(n\) terms is no greater than the absolute value of the first omitted term. Mathematically, this can be described by the error condition: \(|s - s_n| < |b_{n+1}|\).

In our problem, we need this error to be less than 0.001 for accurate approximation. Thus, one solves \(\frac{n+1}{(n+1)^2 + 1} < 0.001\), seeking the smallest \(n\) satisfying this constraint. Subsequently, solving the inequality confirms that using 32 terms ensures the error requirement is met.