Problem 3 - Thomas Calculus

Question

Use the Integral Test to determine if the series in Exercises \(1-12\) converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$ \sum_{n=1}^{\infty} \frac{1}{n^{2}+4} $$

Step-by-Step Solution

Answer

The series \( \sum_{n=1}^{\infty} \frac{1}{n^{2}+4} \) converges by the Integral Test.

Step 1: Identify the Function

The series given is \( \sum_{n=1}^{\infty} \frac{1}{n^{2}+4} \). We identify the function \( f(x) = \frac{1}{x^{2}+4} \).

Step 2: Verify Conditions for the Integral Test

For the Integral Test to apply, the function \( f(x) \) must be positive, continuous, and decreasing for \( x \geq 1 \). The function \( \frac{1}{x^{2}+4} \) is positive and continuous for all \( x \geq 1 \). To check if it's decreasing, we find the derivative and show that it is negative. Calculating the derivative \( f'(x) = -\frac{2x}{(x^{2}+4)^{2}} \), we see that this is negative for \( x \geq 1 \), confirming \( f(x) \) is decreasing.

Step 3: Evaluate the Improper Integral

Now, compute the integral \( \int_{1}^{\infty} \frac{1}{x^{2}+4} \, dx \). Use the substitution \( x = 2\tan(\theta) \), \( dx = 2\sec^{2}(\theta) \, d\theta \), which changes the integral to \( \int \frac{2 \sec^{2}(\theta)}{4\tan^{2}(\theta) + 4} \, d\theta = \int \frac{1}{2} \, d\theta \). Evaluating, this becomes \( \frac{1}{2} \cdot (\theta + C) \). Back-substituting, we find \( \theta = \tan^{-1}(\frac{x}{2}) \).

Step 4: Determine Convergence of the Integral

Calculate the definite integral \( \int_{1}^{\infty} \frac{1}{x^{2}+4} \, dx \) using the result from substitution. This results in \( \left[ \frac{1}{2} \tan^{-1}\left(\frac{x}{2}\right) \right]_{1}^{\infty} \). As \( x \to \infty \), \( \tan^{-1}\left(\frac{x}{2}\right) \to \frac{\pi}{2} \). So the integral evaluates to \( \frac{\pi}{4} - \frac{1}{2}\tan^{-1}(\frac{1}{2}) \), which is a finite number, indicating that the integral converges.

Step 5: Conclude the Series Convergence

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

Key Concepts

Convergence

The concept of convergence in mathematics refers to the notion that a series can approach a specific value or limit as the number of terms goes to infinity. When applied to series, convergence means that as we add more and more terms, the sum of the series gets closer and closer to a specific number. If a series converges, its sum is finite.
Understanding whether a series converges is crucial because it tells us if the infinite sum has a meaningful result.

• For example, the series \( \sum_{n=1}^{\infty} \frac{1}{n^2+4} \) converges, meaning that adding up its infinite terms results in a finite sum.
• Convergence can be checked using various tests, and the Integral Test is one such method.

The Integral Test is particularly useful for determining convergence, especially for series that involve functions we can integrate.

Series

A series is an expression representing the sum of an infinite sequence of terms. It is usually written in the form of an infinite sum \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) are the terms of the series. Series can either converge or diverge depending on their characteristics.
Series are found everywhere in mathematics, physics, and engineering, as they provide a way to sum up an infinite number of factors into a single value.

• An important type of series is the geometric series, which has a common ratio between successive terms.
• Another key type is the harmonic series, which is known to diverge.

In our example, the series \( \sum_{n=1}^{\infty} \frac{1}{n^2+4} \) was determined to be a convergent series through the Integral Test, showing it has a finite sum as we consider more terms.

Improper Integral

Improper integrals are a type of integral where the function is unbounded or the interval of integration is infinite. In calculus, these integrals are handled by taking limits.
An improper integral allows us to compute areas or sums over infinite intervals or areas where the function doesn't behave nicely.

• To evaluate an improper integral, you usually replace the infinite limit with a limit as a variable approaches infinity. This transforms the problem into one that's more manageable.
• In the exercise, the improper integral \( \int_{1}^{\infty} \frac{1}{x^2+4} \, dx \) represents the sum of the infinite series in a continuous form.

Calculating this integral involved substitution, which simplified the expression for ease of evaluation.

Continuous Functions

Continuous functions are functions that have no interruptions, jumps, or breaks in their domain. This means that small changes in the input result in small changes in the output. For a function to be eligible for certain mathematical tests, like the Integral Test, it must be continuous over the range of interest.
Being continuous implies the function smoothly extends across an interval, allowing for integrals to be applied accurately.

• In the solution given, the function \( f(x) = \frac{1}{x^2+4} \) is noted to be continuous for \( x \geq 1 \).
• This continuity combined with positivity and being decreasing are preconditions for the Integral Test.

Ensuring that these conditions are met enables us to use powerful tools like the Integral Test to find the convergence of series.