Question

State the Direct Comparison Test and give an example of its use

Short answer

The Direct Comparison Test is used to determine if a series converges or diverges by comparing it to a series whose converge or divergence is known. In the example, the series \( \sum_{n=1}^{\infty} \frac{1}{2n+1} \) diverges due to the Direct Comparison Test.

Step-by-step solution

Direct Comparison Test Definition

The Direct Comparison Test (also known as direct comparison test for convergence) is used to analyze whether a given series converges or diverges. The test states that if we have two series \( \sum_{n=1}^{\infty} a_n \) and \( \sum_{n=1}^{\infty} b_n\) where all terms are non-negative and \( a_n \leq b_n\) for all n, then if \( \sum_{n=1}^{\infty} b_n \) converges then \( \sum_{n=1}^{\infty} a_n \) also converges. Conversely, if \( \sum_{n=1}^{\infty} a_n \) diverges then \( \sum_{n=1}^{\infty} b_n \) also diverges.

Example of The Direct Comparison Test Usage

Consider the series \( \sum_{n=1}^{\infty} \frac{1}{2n+1} \). We want to find out if it converges or diverges. We compare this to \( \sum_{n=1}^{\infty} \frac{1}{2n} \) which we know is a divergent series. Since \(\frac{1}{2n+1} \geq \frac{1}{2n}\), by Direct Comparison Test, \( \sum_{n=1}^{\infty} \frac{1}{2n+1} \) diverges.