Problem 31 - 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. $$ \sum_{k=1}^{\infty} \frac{k}{1+k^{4}} $$

Step-by-Step Solution

Answer

Using the partial sum with at least 50 terms will keep the error within 0.0002.

Step 1: Analyze the Series

The series given is \( \sum_{k=1}^{\infty} \frac{k}{1+k^{4}} \). This is an alternating series. Alternating series meet certain conditions which allow for estimation of error. Let's first confirm whether this series is alternating.

Step 2: Verify Convergence

We can apply the Alternating Series Test. A series \( \sum_{k=1}^{\infty} (-1)^{k} b_{k} \) converges if \( b_k \) is decreasing and \( \lim_{k \to \infty} b_k = 0 \). Here, it's more complex due to non-negative terms, so consider comparison tests instead. We check convergence since \( \frac{k}{1+k^{4}} \sim \frac{1}{k^3} \), known convergent series.

Step 3: Utilize the Integral Test for Error Approximation

To estimate the error of a convergent series like \( \sum \frac{1}{k^3} \), often close in behavior, we use the next term: the integral test shows \( \int_{n}^{\infty} \frac{dx}{x^3} = \frac{1}{2n^2} \). For \( \frac{1}{2n^2} \leq 0.0002 \), solve \( n^2 \geq 2500 \).

Step 4: Solve for n

Given \( \frac{1}{2n^2} \leq 0.0002 \), \( n^2 \geq 2500 \). Solving as \( n \geq \sqrt{2500} = 50 \). Therefore, \( n \) should be at least 50 to ensure the required error bound of 0.0002.

Key Concepts

Alternating Series

In mathematics, an alternating series is a series where the terms alternate in sign. This means if one term is positive, the next is negative, and vice versa. An example of an alternating series is

• \( rac{1}{1} - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} \cdots \)

Alternating series are important because they have specific conditions that allow us to estimate their error when calculating their sum. A critical condition of an alternating series is that its terms must decrease in absolute value and tend toward zero as the series progresses.
For example, in the series discussed in the exercise, we see

• \( (-1)^k \cdot \frac{k}{1+k^4} \)

While this doesn't strictly follow the standard alternating series format, it provides an insight into scenarios that require careful evaluation of convergence using other methods.

Convergent Series

A series is said to be convergent if its terms approach a certain limit as they are progressively added up. This means that the infinite sum of the series will result in a finite value. In practical terms, convergence tells us that a series "settles down" to a single number.
In the context of the exercise, we explore the series

• \( \sum \frac{k}{1+k^4} \)

To determine if a series converges, various tests can be used such as the Comparison Test or the Alternating Series Test. Although the exercise hinted at using these, the similarity to the

• \( \sum \frac{1}{k^3} \)

convergence pattern simplifies this analysis, using the fact that \( \frac{1}{k^3} \) is known to converge. This means the original series is likely convergent too, based on its comparison behavior.

Integral Test

The Integral Test is a method used to determine the convergence of a series. It is useful when each term of the series can be defined as a continuous, positive, decreasing function. The basic idea is to integrate the underlying function from a specific point to infinity. If this integral converges, so does the series.
In the problem, the Integral Test was subtly applied by evaluating the integral of a function similar to the series in question:

• \( \int_{n}^{\infty} \frac{dx}{x^3} = \frac{1}{2n^2} \)

This provides an error estimation method, allowing us to determine how large \( n \) must be to maintain a desired accuracy (in this case, an error no more than 0.0002). Whenever you're handling series, checking whether the Integral Test can apply is a handy way to determine convergence and develop error estimations.

Partial Sum Approximation

Partial sum approximation refers to evaluating the sum of a series by adding up a finite number of its terms. Instead of summing to infinity, which isn't always possible or practical, we sum up to a specific point \( n \) and use this as an approximation for the complete series.
In the exercise, the partial sum approximation is used to achieve an error margin. We decided on a minimal \( n \) that aligns the remaining terms to behave like \( \frac{1}{2n^2} \), offering a controlled excess of error.
By setting \( \frac{1}{2n^2} \leq 0.0002 \) and solving \( n^2 \geq 2500 \), we calculate that using the first 50 terms results in a satisfactory approximation with the desired accuracy. Partial sums are valuable when a complete series can't be computed, and understanding how large an \( n \) you need can greatly improve the precision of your calculations.