How to Use Comparison test for Convergence

• 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.

The Comparison Test decides a series by bounding it against one whose behavior is known: if a series is term-by-term smaller than a convergent series it converges, and if it is larger than a divergent series it diverges. Below: the conditions, and converging and diverging worked examples.

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.

Frequently Asked Questions

Mastering Grade 3 Math The Ultimate Step by Step Guide to Acing 3rd Grade Math

Download

$29.99 Original price was: $29.99.$16.99Current price is: $16.99.

To effectively prepare your child for their math test, consider using resources that engage and challenge them while reinforcing core concepts. One excellent approach is to introduce them to Top 10 Grade 3 Math Books Inspiring Young Mathematicians To Explore. These books can provide insightful explanations and examples that resonate with young learners. Additionally, supplementing their study with diverse Worksheets can help reinforce the material covered in books and provide practical experience with different types of math problems, aiding in better preparation for their tests.

The Ultimate Elementary School Math Bundle Grade 3-5

Download

$109.99 Original price was: $109.99.$54.99Current price is: $54.99.