Problem 7 - 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. $$ \sum_{n=1}^{\infty} \frac{n}{n^{2}+4} $$

Step-by-Step Solution

Answer

The series diverges.

Step 1: Identify the function

The given series is \( \sum_{n=1}^{\infty} \frac{n}{n^{2}+4} \). For the Integral Test, we need a function \( f(x) = \frac{x}{x^{2}+4} \) that represents the terms of the series.

Step 2: Check the conditions of the Integral Test

The function \( f(x) = \frac{x}{x^{2}+4} \) must be positive, continuous, and decreasing for all \( x \geq 1 \). Evaluating this, \( f(x) > 0 \) and is continuous for \( x \geq 1 \). To check if \( f(x) \) is decreasing, calculate the derivative: \( f'(x) = \frac{d}{dx}\left(\frac{x}{x^2+4}\right) \). By using the quotient rule, \( f'(x) = \frac{(x^2+4)(1) - x(2x)}{(x^2+4)^2} = \frac{4-x^2}{(x^2+4)^2} \). Since \( x^2 \geq 4 \) for \( x \geq 1 \), \( f'(x) \leq 0 \). Therefore, \( f(x) \) is decreasing.

Step 3: Set up the integral

The integral we need to evaluate is \( \int_{1}^{\infty} \frac{x}{x^2 + 4} \, dx \). This integral will help us determine if the series converges or diverges.

Step 4: Compute the integral

To evaluate \( \int \frac{x}{x^2 + 4} \, dx \), use substitution: let \( u = x^2 + 4 \), and \( du = 2x \, dx \) which implies \( \frac{du}{2} = x \, dx \). The integral becomes \( \int \frac{1}{u} \cdot \frac{du}{2} = \frac{1}{2} \int \frac{1}{u} \, du = \frac{1}{2} \ln |u| + C = \frac{1}{2} \ln |x^2 + 4| + C \). Evaluate the improper integral as \( \lim_{b \to \infty} \left( \frac{1}{2} \ln |b^2 + 4| - \frac{1}{2} \ln 5 \right) \). As \( b \to \infty \), \( \ln |b^2 + 4| \to \infty \), hence the integral diverges.

Step 5: Conclude about the series

Since the integral \( \int_{1}^{\infty} \frac{x}{x^2 + 4} \, dx \) diverges, by the Integral Test, the series \( \sum_{n=1}^{\infty} \frac{n}{n^2 + 4} \) also diverges.

Key Concepts

Convergence and Divergence

Understanding convergence and divergence is essential when analyzing infinite series. When we talk about convergence, we mean that the sum of the series approaches a specific value as the number of terms goes to infinity. On the other hand, divergence occurs when the series does not settle to any particular value, continuing to increase or decrease indefinitely.
To determine if a series converges or diverges, mathematicians use various tests. One common method is the Integral Test, which involves checking if the corresponding function’s integral converges or diverges.

• If the integral is finite, the series converges.
• If the integral is infinite, the series diverges.

It's crucial to recognize that not all series can be analyzed with the Integral Test, as certain conditions have to be satisfied. This involves ensuring the function is positive, continuous, and decreasing from a particular starting point. In our example, the series diverged because the integral did not converge to a finite number, implying that the sum of the series also doesn't settle at a specific value.

Improper Integrals

Improper integrals can be a complex topic, but grasping the basic idea helps in understanding the Integral Test. An integral is termed 'improper' when either the limits of integration are infinite or the function being integrated becomes infinite within the integration limits.
To compute an improper integral, we usually employ limits to handle the infinities. In our case, the integral \[\int_{1}^{\infty} \frac{x}{x^2 + 4} \, dx\] involves the upper limit going to infinity. By evaluating the limit, the result shows whether the integral converges or diverges.
Approaching improper integrals step-by-step:

• Substitute or manipulate the integral when necessary for easier calculation.
• Assess convergence by evaluating limits – if the resulting limit is finite, the integral converges.
• If the limit approaches infinity, the integral diverges.

This process can reveal properties of the related series, and in our scenario, the divergence of the integral reflected that the series itself diverged.

Differentiation

Differentiation plays a pivotal role when analyzing functions for the Integral Test. We must confirm that the function representing the terms of the series is decreasing from a given point onward. To do this, we take the derivative of the function.
The derivative helps us see how the function behaves over its domain:

• If the derivative is negative for a range of values, the function decreases.
• If it’s positive, the function increases.

Let's take a look at the function \( f(x) = \frac{x}{x^2+4} \) from our exercise. We found the derivative to be:
\[ f'(x) = \frac{4-x^2}{(x^2+4)^2} \]
For all \( x \geq 1 \), since \( x^2 \) grows more rapidly than 4, \( f'(x) \leq 0 \). This means \( f(x) \) is indeed decreasing, satisfying one of the key conditions for applying the Integral Test.
Understanding differentiation isn’t just about calculating derivatives; it’s about interpreting them in the context of broader mathematical problems. In this way, differentiation aids in determining the validity of methods like the Integral Test.