📘 Chapter 12: Convergence & Divergence of Infinite Series
🔁 What Is an Infinite Series?
An infinite series is the sum of the terms of an infinite sequence:
\[ \sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \cdots \]
- If the sum approaches a finite value, the series converges.
- If it grows without bound or oscillates, the series diverges.
✅ Convergence vs ❌ Divergence
| Concept | Description |
|---|---|
| Convergent Series | The partial sums approach a finite limit. Example: \( \sum \frac{1}{n^2} \) |
| Divergent Series | The partial sums do not approach a finite limit. Example: \( \sum \frac{1}{n} \) |
🔍 Tests for Convergence
1. Nth-Term Test for Divergence
- If \( \lim_{n \to \infty} a_n \neq 0 \), the series diverges.
- If \( \lim_{n \to \infty} a_n = 0 \), the test is inconclusive.
2. Geometric Series Test
- Series of the form \( \sum ar^n \)
- Converges if \( |r| < 1 \); diverges if \( |r| \geq 1 \)
3. p-Series Test
- Series of the form \( \sum \frac{1}{n^p} \)
- Converges if \( p > 1 \); diverges if \( p \leq 1 \)
4. Integral Test
If \( f(n) = a_n \) is positive, continuous, and decreasing:
\[ \sum a_n \text{ and } \int f(x)\,dx \text{ both converge or both diverge} \]
5. Comparison Test
- If \( 0 \leq a_n \leq b_n \) and \( \sum b_n \) converges, then \( \sum a_n \) converges.
- If \( a_n \geq b_n \geq 0 \) and \( \sum b_n \) diverges, then \( \sum a_n \) diverges.
6. Limit Comparison Test
If \( \lim_{n \to \infty} \frac{a_n}{b_n} = c \) where \( c > 0 \), then both series behave the same.
7. Ratio Test
Compute \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \)
- If \( L < 1 \), converges
- If \( L > 1 \), diverges
- If \( L = 1 \), inconclusive
8. Root Test
Compute \( L = \lim_{n \to \infty} \sqrt[n]{|a_n|} \)
- Same conclusions as the Ratio Test
📊 Examples
Example 1: \( \sum \frac{1}{n^2} \)
This is a p-series with \( p = 2 > 1 \) → ✅ Converges
Example 2: \( \sum \frac{1}{n} \)
This is a p-series with \( p = 1 \) → ❌ Diverges (harmonic series)
📝 Practice
Q: Determine whether the series \( \sum \frac{n}{n^2 + 1} \) converges or diverges using the Limit Comparison Test.