Series Convergence Tests
Key tests for determining if an infinite series converges or diverges.
Ratio test, comparison test, integral test with worked examples.
Geometric Series Test
The geometric series is the simplest and most commonly tested series. If the common ratio r has an absolute value less than 1, the series converges to a finite sum. If |r| ≥ 1, the series diverges.
Ratio Test
If L < 1, the series converges absolutely. If L > 1, the series diverges. If L = 1, the test is inconclusive and you need a different test.
The ratio test works especially well for series involving factorials and exponentials.
p-Series Test
The harmonic series (p = 1) diverges, but 1/n² (p = 2) converges. This is a quick test when you recognize a series as a p-series.
Comparison Test
Compare your series to a known series:
- If 0 ≤ an ≤ bn and Σbn converges, then Σan converges
- If an ≥ bn ≥ 0 and Σbn diverges, then Σan diverges
Variables
| Symbol | Meaning |
|---|---|
| an | The nth term of the series |
| r | Common ratio in a geometric series |
| p | Exponent in a p-series (Σ 1/np) |
| L | Limit from the ratio test |
| n | Index variable (positive integer) |
Example 1: Ratio Test
Does the series Σ (n! / 3n) converge or diverge? (n = 1 to ∞)
Apply the ratio test: L = lim |an+1 / an|
an = n! / 3n, so an+1 = (n+1)! / 3n+1
L = lim |(n+1)! / 3n+1 × 3n / n!| = lim |(n+1) / 3|
As n → ∞, (n+1)/3 → ∞
L = ∞ > 1, so the series diverges. (Factorials grow much faster than exponentials.)
Example 2: Geometric Series
Find the sum of the series: 1 + 1/3 + 1/9 + 1/27 + ... (a geometric series with a = 1, r = 1/3)
Check convergence: |r| = |1/3| = 1/3 < 1, so the series converges
Apply the sum formula: S = a / (1 − r) = 1 / (1 − 1/3) = 1 / (2/3)
S = 3/2 = 1.5
Which Test to Use
Choosing the right convergence test is key to solving series problems efficiently.
- Geometric series: Use when the series has the form Σarn
- p-series test: Use when the series looks like Σ1/np
- Ratio test: Use when terms involve factorials, exponentials, or powers of n
- Comparison test: Use when you can bound the series by a simpler known series
- Alternating series test: Use when signs alternate (+, −, +, −, ...)