How to Use Comparison test for Convergence

The Comparison Test is a method used to determine the convergence or divergence of an infinite series by comparing it with another series whose convergence properties are known. For a given series \(\sum a_n\), the test compares it to another series \(\sum b_n\), where \(b_n\) has simpler terms, making it easier to analyze.

The test works as follows:

• If \(0≤an≤bn\)​ for all \(n\) large enough and \(\sum b_n\) converges, then \(\sum a_n\) also converges.
• If \(an​≥bn​\) and \(\sum b_n\) diverges, then \(\sum a_n\) also diverges.

This test is helpful when it’s difficult to directly assess the convergence of the original series. By finding a suitable comparison series that converges or diverges, you can quickly conclude the behavior of the given series. The Comparison Test is particularly useful for series involving polynomial, exponential, or logarithmic functions.

Here are two examples demonstrating the Comparison Test:

1. Converging Series Example:

Series: \(\sum_{n=1}^{\infty} \frac{2}{n^3 + 1}\)

Comparison Series: \(\sum_{n=1}^{\infty} \frac{1}{n^3}\)

• The terms \(\frac{2}{n^3 + 1}\) are smaller than \(\frac{2}{n^3}\) for all \(n \geq 1\), and we know that \(\sum \frac{1}{n^3}\) converges (since it’s a p-series with \(p = 3\)).
• By the Comparison Test, since the comparison series converges, the original series also converges.

2. Diverging Series Example:

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

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

• The series \(\frac{1}{n \ln(n)}\) is larger than \(\frac{1}{n^2}\) for large \(n\), and we know that the harmonic series \(\sum \frac{1}{n}\) diverges.
• By the Comparison Test, since the comparison series diverges, the original series diverges as well.