Problem 3 - 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. $$ \sum_{n=1}^{\infty} \frac{1}{n^{2}+4} $$

Step-by-Step Solution

Answer

The series converges by the Integral Test.

Step 1: Identify function for Integral Test

First, identify the function corresponding to the series terms. The series \( \sum_{n=1}^{\infty} \frac{1}{n^2 + 4} \) corresponds to the function \( f(x) = \frac{1}{x^2 + 4} \).

Step 2: Check conditions for Integral Test

Ensure that the function \( f(x) = \frac{1}{x^2 + 4} \) is positive, continuous, and decreasing for \( x \geq 1 \). We know that \( f(x) > 0 \) and it is continuous for all \( x > 0 \), particularly for \( x \geq 1 \). To check if it is decreasing, compute the derivative and ensure it is non-positive.

Step 3: Compute the derivative to check decreasing nature

Calculate the derivative of \( f(x) = \frac{1}{x^2 + 4} \). Using the quotient rule, we find \( f'(x) = -\frac{2x}{(x^2 + 4)^2} \). Since \( -\frac{2x}{(x^2 + 4)^2} \leq 0 \) for \( x \geq 1 \), the function is decreasing.

Step 4: Set up the integral

The Integral Test involves evaluating the improper integral \( \int_{1}^{\infty} \frac{1}{x^2 + 4} \, dx \). If this integral converges, so does the series, and vice versa.

Step 5: Evaluate the improper integral

To solve \( \int_{1}^{\infty} \frac{1}{x^2 + 4} \, dx \), recognize it as an arctangent integral. It evaluates to \( \frac{1}{2} \left[ \arctan\left(\frac{x}{2}\right) \right]_1^{\infty} \), which simplifies to \( \frac{\pi}{4} - \frac{1}{2} \arctan\left(\frac{1}{2}\right) \). This converges since both terms are finite.

Step 6: Conclude using the Integral Test

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

Series Convergence

When tackling problems related to the convergence or divergence of a series, the Integral Test is a useful tool. In the realm of series, convergence means that as you keep adding more and more terms, the sum approaches a finite value. For the series \( \sum_{n=1}^{\infty} \frac{1}{n^{2}+4} \), we use the corresponding continuous function \( f(x) = \frac{1}{x^2 + 4} \) to help analyze its behavior.
By determining if this function starts positive, stays continuous, and decreases as \( x \) increases, we satisfy the basic conditions required for the Integral Test.

• The function must be positive because each term of the series is positive.
• Continuity is crucial, as a break can lead to unpredictable sums.
• Lastly, the function should symmetrically decrease, ensuring a downward sloping trend.

Once these conditions hold, we assess the convergence by comparing it with a related improper integral, providing solid insight into the behavior of the original series.

Improper Integrals

An improper integral extends the concept of definite integrals to cases where the interval of integration is infinite, or where the function has undefined areas within the range. For an integral like \( \int_{1}^{\infty} \frac{1}{x^2 + 4} \, dx \), the range of integration could infinitely extend to include all values starting from 1 onwards, thereby termed as an improper integral.

• In practice, such integrals assist in determining the ultimate behavior of series through the Integral Test.
• To evaluate an improper integral, break it down into simpler functions with known integral results, such as recognizing \( \frac{1}{x^2 + 4} \) as a derivation of the arctangent function.
• The solution, obtained through calculated limits, informs us if the series converges based on whether the integral result is finite or infinite.

For instance, in this exercise, the computation leads us to a conclusion that the original series converges since the evaluated integral is finite, equating to \( \frac{\pi}{4} - \frac{1}{2} \arctan\left(\frac{1}{2}\right) \). Therefore, improper integrals serve as a powerful indicator of infinite series behavior.

Derivative Calculation

To determine if the expression \( \frac{1}{x^2 + 4} \) consistently decreases, calculating its derivative is essential. This step ensures the suitability of the function for the Integral Test, impacting the decision on convergence.
We'll use the quotient rule, a calculus method used when calculating the derivative of a ratio of two functions. This is vital when the function isn't a simple polynomial.

• The function's derivative \( f'(x) = -\frac{2x}{(x^2 + 4)^2} \) was computed through the quotient rule.
• Given \( f'(x) \leq 0 \) for \( x \geq 1 \), it demonstrates that the function decreases consistently.
• A decreasing function suggests that the elements of the series diminish, which is supportive of a converging sum.

Understanding derivative calculations helps confirm the conditions needed for the Integral Test, providing a solid foundation towards solving the larger problem of series convergence.