Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

For the series given, determine how large \(n\) must be so that using the nth partial sum to approximate the series gives an error of no more than 0.0002. $$ \sum_{k=1}^{\infty} \frac{1}{k^{3}} $$

Step-by-Step Solution

Answer

The smallest integer \( n \) needed is 70.

Step 1: Understanding the Series

The series given is \( \sum_{k=1}^{\infty} \frac{1}{k^{3}} \), which is a p-series with \( p = 3 \). Since \( p > 1 \), the series converges.

Step 2: Determining the Error Formula

To determine the error when approximating an infinite series by its nth partial sum, \( S_n \), we use the error bound for a convergent p-series: \( R_n = \left| S - S_n \right| < \frac{1}{(n+1)^2} \).

Step 3: Setting Up the Inequality

We want the error \( R_n \) to be less than or equal to 0.0002. Therefore, we set up the inequality: \( \frac{1}{(n+1)^2} < 0.0002 \).

Step 4: Solving the Inequality for \( n \)

Rearrange the inequality: \( (n+1)^2 > 5000 \). Taking the square root gives \( n+1 > \sqrt{5000} \).

Step 5: Calculating \( n \)

Compute \( \sqrt{5000} \) which is approximately 70.71. Thus, \( n + 1 > 70.71 \) implies \( n > 69.71 \).

Step 6: Determining the Smallest Integer \( n \)

Since \( n \) must be an integer, we take the smallest integer greater than 69.71, which is \( n = 70 \).

Key Concepts

Understanding Error Approximation in Series

Error approximation is a crucial concept when working with infinite series. It revolves around estimating the difference between the actual sum of the series, denoted as \( S \), and an approximate sum obtained through calculating a finite number of initial terms, named the nth partial sum, \( S_n \). This difference, or error, is referred to as \( R_n = |S - S_n| \).

For a convergent p-series, like \( \sum_{k=1}^{\infty} \frac{1}{k^3} \), there's a practical way to estimate this error. The error bound can be computed using the formula \( R_n < \frac{1}{(n+1)^2} \), allowing us to understand how far off our approximation could be. For example, if one desires an error not exceeding 0.0002, this formulation helps set the criteria for choosing \( n \). Understanding this process is pivotal in ensuring precision and accuracy when examining infinite series.

The Concept of Infinite Series

An infinite series is a sum of an infinite sequence of terms. In mathematics, this concept is fundamental as it allows for expressing functions as sums of infinitely many terms, which are usually ordered according to a specific rule or formula.

Consider the series \( \sum_{k=1}^{\infty} \frac{1}{k^3} \), which is an example of a p-series where each term diminishes as \( k \) increases. In infinite series, we often categorize them by their potential to sum to a finite number (convergence) or to not have a sum (divergence).

Key points about infinite series include:

• Each sequence's behavior is dictated by its general term \( a_k \).
• Series can be convergent or divergent based on their terms' limit as \( k \) approaches infinity.
• Understanding the properties of series facilitates their application in mathematical and practical scenarios.

Understanding Partial Sums

A partial sum is the sum of the first \( n \) terms of an infinite series, denoted as \( S_n \). This tool is essential for approximating the total sum of an infinite series since calculating all terms from an infinite series explicitly is impossible.

Partial sums offer a way to simplify complex computations:

• By evaluating \( S_n \), you get an approximate value of the series up to the nth term.
• The partial sum provides insight into the series' behavior and convergence.
• For the series \( \sum_{k=1}^{\infty} \frac{1}{k^3} \), the nth partial sum helps assess how close the approximation is to the real series sum.

Utilizing partial sums is a stepping stone to deeper analysis, such as finding where an acceptable error threshold lies, ensuring appropriate accuracy in applications.

Applying Convergence Criterion

The convergence criterion is a rule that helps decide if an infinite series converges or diverges. For instance, a p-series \( \sum_{k=1}^{\infty} \frac{1}{k^p} \) converges if the exponent \( p > 1 \). This criterion is foundational in determining the limits and practicality of using series in calculations.

Using convergence criteria:

• Helps identify if the series will have a finite sum.
• Ensures you apply the series appropriately in mathematical models, knowing it reaches practical outcomes.
• For the series \( \sum_{k=1}^{\infty} \frac{1}{k^3} \), since \( p = 3 \) which is greater than 1, the series is confirmed to converge.

Recognizing convergence assists in setting up precise calculations, like evaluating how many terms are needed for a predetermined acceptable error margin, creating robust models for problem-solving.