Problem 82 - Essential Calculus

Question

State the Integral Test and give an example of its use.

Step-by-Step Solution

Answer

The Integral Test can be stated as follows: 'Given a function \( f \) that is continuous, positive, and decreasing over the interval \([k, +\infty) \), for some positive integer \( k \) the infinite series \( \sum_{n=k}^{+\infty} f(n) \) and the improper integral \( \int_{k}^{+\infty} f(x) dx \) either both converge or both diverge.' An example of its application would be to test the given series \(\sum_{n=2}^{+\infty} \frac{1}{n \log n} \) that is observed to be positive, decreasing and continuous. Applying the Integral Test, by evaluating the integral \(\int_{2}^{+\infty} \frac{1}{x \log x } dx \), results in an infinite value which implies that both the integral and the series diverge.

Step 1: State the Integral Test

The Integral Test is a test of theories in mathematics, used in calculus to test the convergence of infinite series. It is stated as follows: Given a function \( f \) that is continuous, positive, and decreasing over the interval \([k, +\infty) \), for some positive integer \( k \) the infinite series \( \sum_{n=k}^{+\infty} f(n) \) and the improper integral \( \int_{k}^{+\infty} f(x) dx \) either both converge or both diverge.

Step 2: Example of the Integral Test

Let's consider the infinite series \(\sum_{n=2}^{+\infty} \frac{1}{n \log n} \) to be tested for convergence or divergence. We observe that the series is positive, decreasing and continuous so the conditions of the Integral Test are verified. Therefore, we can apply the integral test by evaluating the integral: \(\int_{2}^{+\infty} \frac{1}{x \log x } dx \).

Step 3: Evaluate the Integral

Firstly, set \( u = \log x \) and \( dv = \frac{1}{x} dx \). Thus, by parts \( \int u dv = uv - \int v du \), the integral becomes: \(\int_{\log 2}^{+\infty} 1 du = [u]_{\log 2}^{+\infty} = +\infty - \log 2 \), which is equal to infinity. Therefore, the integral diverges.