Understanding the Integral Test for Series Convergence

The Integral Test is used to determine the convergence or divergence of an infinite series by comparing it to an improper integral. If you have a series \(\sum a_n\), where \(a_n = f(n)\) and \(f(x)\) is a continuous, positive, and decreasing function for \(\geq 1\), you can apply the Integral Test. The test compares the series to the improper integral of \(f(x)\) over the interval \([1, \infty)\): \(\int_1^\infty f(x) \, dx\)

• If the integral converges, the series also converges.
• If the integral diverges, the series also diverges.

The test is particularly useful for series where the terms resemble well-known functions that are easy to integrate, such as exponential or polynomial terms. However, if the function doesn’t meet the criteria of being positive, continuous, and decreasing, the test cannot be applied.

Here are two more examples demonstrating the Integral Test:

1. Converging Example:

Series: \(\sum_{n=1}^{\infty} \frac{e^{-n}}{n}\)

Function: \(f(x) = \frac{e^{-x}}{x}\)

Apply the Integral Test: Evaluate the integral: \(\int_1^\infty \frac{e^{-x}}{x} \, dx\)

This integral converges because the function \(\frac{e^{-x}}{x}\) decays rapidly for large \(x\). Therefore, the series \(\sum_{n=1}^{\infty} \frac{e^{-n}}{n}\) converges.

2. Diverging Example:

Series: \(\sum_{n=1}^{\infty} \frac{1}{n \ln(n)}\)

Function: \(f(x) = \frac{1}{x \ln(x)}\)

Apply the Integral Test: Evaluate the integral: \(\int_2^\infty \frac{1}{x \ln(x)} \, dx\)

This is a known divergent integral (it diverges logarithmically). Therefore, the series \(\sum_{n=1}^{\infty} \frac{1}{n \ln(n)}\) diverges.