Series & Power Series Survival Guide
1. The Strategy Tree (Which Test?)
| If the series looks like... |
Use this Test |
What to look for: |
| 1 / np |
p-Series |
Convergent if p > 1. Divergent if p ≤ 1. |
| arn-1 or (constant)n |
Geometric |
Conv. if |r| < 1. Sum = First / (1 - r). |
| Factorials (n!) or nn |
Ratio Test |
Limit L < 1 (Conv), L > 1 (Div), L = 1 (Fail). |
| (-1)n bn |
Alternating (AST) |
Terms must decrease to 0. |
| Polynomials (n2+1)/(n3-n) |
Limit Comparison |
Compare to the highest powers. |
2. Finding the Interval of Convergence
Step 1: Apply the Ratio Test to the absolute value of the terms.
Step 2: Solve the inequality Limit < 1 to find the Radius (R).
Step 3: Check the endpoints of your interval individually! This is where most people lose marks. Plug them back into the original series and test for convergence.
3. Essential Maclaurin Series (Memorize!)
Exponential: ex = Σ (xn / n!) = 1 + x + x2/2! + x3/3! + ...
Sine: sin x = Σ [(-1)n x2n+1 / (2n+1)!] = x - x3/3! + x5/5! - ...
Cosine: cos x = Σ [(-1)n x2n / (2n)!] = 1 - x2/2! + x4/4! - ...
Geometric: 1 / (1 - x) = Σ xn = 1 + x + x2 + ... (|x| < 1)
Inverse Tan: arctan x = Σ [(-1)n x2n+1 / (2n+1)] = x - x3/3 + x5/5 - ...
4. Quick Reminders
- The Harmonic Series (Σ 1/n) always diverges.
- If the limit of the terms isn't 0, the series diverges immediately (Test for Divergence).
- Absolute Convergence → Conditional Convergence. (If it's absolute, it's already conditional).