Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use the Integral Test to determine the convergence or divergence of each of the following series. $$ \sum_{k=2}^{\infty} \frac{7}{4 k+2} $$

Step-by-Step Solution

Answer

The series diverges by the Integral Test.

Step 1: Define the related function

Rewrite the series as a function that can be integrated: \( f(x) = \frac{7}{4x + 2} \). We want to integrate from 2 to infinity to determine the convergence of the series.

Step 2: Check the function conditions

Verify that \( f(x) = \frac{7}{4x + 2} \) is positive, continuous, and decreasing for \( x \geq 2 \). All three conditions hold because: (1) it's positive for \( x \geq 2 \), (2) it's continuous since the denominator never equals zero, and (3) it's decreasing as the derivative \( f'(x) = \frac{-28}{(4x + 2)^2} \) is negative.

Step 3: Set up the improper integral

The integral we need to evaluate is \( \int_{2}^{\infty} \frac{7}{4x + 2} \, dx \). Set this as \( \lim_{b \to \infty} \int_{2}^{b} \frac{7}{4x + 2} \, dx \).

Step 4: Integrate the function

To integrate \( \int \frac{7}{4x + 2} \, dx \), let \( u = 4x + 2 \) so \( du = 4 \, dx \) or \( dx = \frac{du}{4} \). Substitute into the integral: \( \int \frac{7}{u} \cdot \frac{du}{4} \) which simplifies to \( \frac{7}{4} \int \frac{1}{u} \, du \). This integrates to \( \frac{7}{4} \ln|u| + C \). Substitute back for \( x \): \( \frac{7}{4} \ln|4x + 2| + C \).

Step 5: Evaluate the definite integral

Evaluate \( \lim_{b \to \infty} \left[ \frac{7}{4} \ln|4x + 2| \right]_{2}^{b} \). Calculate \( \frac{7}{4} \ln|4b + 2| - \frac{7}{4} \ln|8| \). As \( b \to \infty \), \( \ln(4b + 2) \to \infty \). Thus the integral diverges.

Step 6: Conclusion on series

Since the integral \( \int_{2}^{\infty} \frac{7}{4x + 2} \, dx \) diverges, by the Integral Test, the series \( \sum_{k=2}^{\infty} \frac{7}{4k+2} \) also diverges.

Key Concepts

Convergence and Divergence

When dealing with infinite series, determining whether a series "converges" or "diverges" is important. A series converges if the sum of its infinitely many terms results in a finite value. Conversely, a series diverges if this sum doesn't settle on a single number, meaning it grows endlessly.

In the context of the Integral Test, we compare a series with an improper integral. If the integral of the related function converges to a finite value, the series also converges. Conversely, if the integral diverges, the series does too. For example, in our problem, we found the related integral diverged. Thus, the series itself diverges. This gives us an efficient means to assess the behavior of more complex series.

Here’s a handy approach:

• Define the function that represents the series’ term.
• Verify the conditions of being positive, continuous, and decreasing for the integral test.
• Use the integral to determine convergence or divergence.

Improper Integrals

Improper integrals often appear when we evaluate series through the Integral Test. Unlike standard definite integrals that have bounds set at fixed numbers, improper integrals extend to infinity. This involves more intricate calculations to determine convergence or divergence.

For instance, in this exercise, the integral from 2 to infinity of the function \( f(x) = \frac{7}{4x + 2} \) was calculated. We set it up as a limit: \( \lim_{b \to \infty} \int_2^b \frac{7}{4x + 2} \, dx \). The nature of improper integrals requires evaluating this limit to see where it trends as \( b \) heads to infinity.

Here’s how it’s tackled:

• Recognize the integral's bounds which extend infinitely.
• Set up the integral as a limit where one bound approaches infinity.
• Calculate the indefinite integral, then consider the limit of the expression when substituted back in.

In the provided problem, the limit leads to infinity, indicating divergence. Thus, reflecting on improper integrals presents insights into the behavior of the series at hand.

Calculus Series

Series in calculus consist of the sum of terms following a specific sequence. When the focus is on infinite series, understanding their convergence or divergence becomes crucial. A prominent tool in this analysis is the Integral Test, primarily used in determining when infinite series align with our expectations.

A key element is defining a related continuous function that mirrors the series, hence why the convergence of the integral provides an introspective view into the convergence of the series.

For calculus series, keep in mind:

• Series consist of terms that stem from functions.
• The Integral Test utilizes improper integrals to judge convergence.
• It requires meticulously checking for specific function behavior: positivity, continuity, and monotonic decreasing nature.

In practical calculations, as demonstrated, the Integral Test offers a structured approach by allowing integration to verify the series' fate. Logic behind such series can guide you through many similar problems encountered in calculus.