Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that each alternating series converges, and then estimate the error made by using the partial sum \(S_{9}\) as an approximation to the sum \(S\) of the series (see Examples \(1-3)\). $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{2}{3 n+1} $$

Step-by-Step Solution

Answer

The series converges and the approximation error with \(S_9\) is \(|S - S_9| \leq \frac{2}{31}\).

Step 1: Verify Alternating Series Test Conditions

To apply the Alternating Series Test (AST), ensure that the series \( \sum_{n=1}^{\infty} (-1)^{n+1} a_n \) meets the following criteria:1. The terms \( a_n = \frac{2}{3n+1} \) are positive, i.e., \( a_n > 0 \) for all \( n \).2. \( a_{n+1} \leq a_n \); the sequence is non-increasing.3. \( \lim_{n \to \infty} a_n = 0 \).For our series, each of these criteria can be verified. Since \( \frac{2}{3n+1} \) decreases as \( n \) increases and approaches 0, the series satisfies the AST criteria.

Step 2: Apply the Alternating Series Test

Since our series satisfies the conditions of the Alternating Series Test, we conclude that the series \( \sum_{n=1}^{\infty} (-1)^{n+1} \frac{2}{3n+1} \) is convergent.

Step 3: Determine Partial Sum \(S_9\)

The partial sum \( S_9 \) is the sum of the first 9 terms of the series:\[S_9 = \sum_{n=1}^{9} (-1)^{n+1} \frac{2}{3n+1} = \frac{2}{4} - \frac{2}{7} + \frac{2}{10} - \frac{2}{13} + \frac{2}{16} - \frac{2}{19} + \frac{2}{22} - \frac{2}{25} + \frac{2}{28}\].Calculate each term and sum them to obtain \( S_9 \).

Step 4: Estimate the Error

The error in the approximation of the sum by the partial sum \(S_9\), for an alternating series, is less than or equal to the absolute value of the first omitted term, \(a_{10}\).Calculate \(a_{10} = \frac{2}{3 \times 10 + 1} = \frac{2}{31}\).Thus, the error \(|S - S_9| \leq \frac{2}{31} \). This gives an idea of how accurate our approximation \(S_9\) is to the actual sum \(S\).

Key Concepts

Convergence

When working with an alternating series, one main focus is to determine if the series converges. An alternating series consists of terms that switch signs from positive to negative with each subsequent term. One powerful tool for determining the convergence of such series is the Alternating Series Test (AST).

To apply the AST, the series must meet three key conditions:

• The individual terms, denoted as \(a_n\), must always be positive. This means \(a_n > 0\) for all values of \(n\).
• The terms must be non-increasing. In simple terms, each term \(a_{n+1}\) should be less than or equal to the preceding term \(a_n\). This requirement ensures the sequence is either constant or getting smaller.
• Finally, the terms must approach zero as \(n\) approaches infinity. Mathematically, this is expressed as \(\lim_{n \to \infty} a_n = 0\).

For the series \(\sum_{n=1}^{\infty} (-1)^{n+1} \frac{2}{3n+1}\), all these conditions are satisfied. The terms \(\frac{2}{3n+1}\) are positive, they decrease as \(n\) increases, and they approach zero. Therefore, the AST confirms that this alternating series converges.

Partial Sum

Once we establish the convergence of a series, the next step often involves calculating partial sums. A partial sum, like \(S_9\), is the sum of a finite number of terms from the series. In this case, \(S_9\) represents the sum of the first nine terms of our given alternating series.

To compute \(S_9\), we simply follow the pattern of adding and subtracting the series terms as noted: \[S_9 = \frac{2}{4} - \frac{2}{7} + \frac{2}{10} - \frac{2}{13} + \frac{2}{16} - \frac{2}{19} + \frac{2}{22} - \frac{2}{25} + \frac{2}{28}\].

Calculate each of these fractions and then combine them, respecting the alternating signs, to find the numerical value of \(S_9\). This partial sum is an approximation of the infinite series and gives us an insight into its value up to the ninth term.

Error Estimation

Estimating the error of an approximation is crucial, especially when dealing with partial sums. In alternating series, the error from truncating a series at a certain point can be effectively estimated. The Alternating Series Error Estimation tells us that the error from using a partial sum, \(S_n\), is less than or equal to the absolute value of the first omitted term.

For our series and using the partial sum \(S_9\), the error will be approximately equal to the term \(a_{10}\), the first term not included in the sum. Here, \(a_{10} = \frac{2}{31} \).

This means that the difference between the actual sum of the infinite series and our partial sum \(S_9\) can be estimated as: \[|S - S_9| \leq \frac{2}{31}\.\]

As a result, this error estimation not only tells us how close our partial sum is to the true value of the series but also provides a measure of accuracy for practical calculations.