Problem 4 - Thomas Calculus

Question

Use the Integral Test to determine if the series in Exercises \(1-10\) converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$ \sum_{n=1}^{\infty} \frac{1}{n+4} $$

Step-by-Step Solution

Answer

The series diverges.

Step 1: Verify Decreasing and Positive Function

The terms of the series given are \( a_n = \frac{1}{n+4} \). First, we need to ensure that \( f(x) = \frac{1}{x+4} \) is continuous, positive, and decreasing for \( x \geq 1 \). Since the function is a rational function with no discontinuity for \( x \geq 1 \), it is continuous and positive. To check if it is decreasing, we observe that \( f'(x) = -\frac{1}{(x+4)^2} \) is negative, indicating the function is decreasing.

Step 2: Set Up the Integral

To use the Integral Test, consider the integral \( \int_{1}^{\infty} \frac{1}{x+4} \, dx \). This integral will help us determine if the series converges or diverges.

Step 3: Evaluate the Integral

Evaluate the integral \( \int_{1}^{\infty} \frac{1}{x+4} \, dx \):\[\int \frac{1}{x+4} \, dx = \ln|x+4| + C\]Evaluate this from 1 to \( \infty \):\[\lim_{b \to \infty} \left( \ln|b+4| \right) - \ln|1+4| = \lim_{b \to \infty} \ln(b+4) - \ln(5)\]As \( b \to \infty \), \( \ln(b+4) \to \infty \). Therefore, the integral diverges.

Step 4: Conclude based on the Integral Test

Since the integral \( \int_{1}^{\infty} \frac{1}{x+4} \, dx \) is divergent, by the Integral Test, the original series \( \sum_{n=1}^{\infty} \frac{1}{n+4} \) also diverges.

Key Concepts

Series Convergence

In the world of calculus, understanding whether a series converges is essential for determining if the sum of its infinite terms reaches a finite limit. Series convergence occurs when adding the terms of a series infinitely results in a finite number. There are different types of tests to determine convergence, and one common method is the Integral Test.

For a series to be convergent through the Integral Test, the function must be positive, continuous, and decreasing. In our exercise, the function given is \( f(x) = \frac{1}{x+4} \). This function meets all the criteria:

• It is positive for \( x \geq 1 \), ensuring it never reaches zero or negative values.
• It's continuous as there are no breaks or spikes in the function graph for the specified domain.
• Lastly, the function is decreasing, as shown by its derivative \( f'(x) = -\frac{1}{(x+4)^2} \), which is always negative.

By confirming these conditions are met, we can confidently apply the Integral Test to determine convergence.

Divergent Series

When series fail to converge, we label them as divergent. A series is divergent when the sum of its terms tends to infinity as more terms are added. This happens because the integral of the function representing the series does not settle to a finite number.

In our exercise, after setting up the integral \( \int_{1}^{\infty} \frac{1}{x+4} \, dx \), we evaluate it from 1 to infinity. The integral simplifies to \( \ln|x+4| \) evaluated from 1 to infinity. This becomes \( \lim_{b \to \infty} \ln(b+4) - \ln(5) \). As \( b \to \infty \), \( \ln(b+4) \to \infty \), indicating the integral diverges. Consequently, the series \( \sum_{n=1}^{\infty} \frac{1}{n+4} \) is divergent, confirming our initial result through the Integral Test.

Calculus

Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. It is often divided into two main parts: differential calculus and integral calculus. While differential calculus focuses on rates of change and slopes of curves, integral calculus is concerned with the accumulation of quantities and areas under curves.

The problem at hand uses integral calculus to determine whether an infinite series converges or diverges. Specific to our exercise, we leverage the Integral Test from integral calculus. This test requires computing improper integrals, which are integrals with infinite limits. It serves as a bridge that connects calculus concepts to series evaluation.

Understanding and applying calculus concepts like the Integral Test is crucial. It provides profound insights into how infinite processes can have tangible, finite results, or in the case of divergent series, how such processes can extend indefinitely without converging. These principles form the foundations for handling complex mathematical analysis and real-world applications, from physics to engineering.