Calc 2: Series Convergence/Divergence FlashCards

Test for Divergence - When to use

If lim(n→∞) aₙ ≠ 0 or the limit doesn't exist, the series diverges. If lim(n→∞) aₙ = 0, the test is INCONCLUSIVE (series might converge or diverge).

Alternating Series Test - When to use

Use when the series alternates between positive and negative terms (has (-1)ⁿ or (-1)ⁿ⁺¹). Check: (1) terms decrease in absolute value: bₙ₊₁ ≤ bₙ, and (2) lim(n→∞) bₙ = 0. If both true, series converges.

Alternating Series Estimation Theorem - What it tells you

For alternating series that converge, the error |s - sₙ| ≤ bₙ₊₁ (the first neglected term). Use this to find how many terms needed for desired accuracy.

Absolute Convergence - Definition

A series Σaₙ is absolutely convergent if Σ|aₙ| converges. KEY: If absolutely convergent, then the series converges!

Conditional Convergence - Definition

A series is conditionally convergent if Σaₙ converges BUT Σ|aₙ| diverges. Example: alternating harmonic series Σ(-1)ⁿ⁺¹/n

Ratio Test - When to use

Best for series with FACTORIALS (n!) or EXPONENTIALS (aⁿ). Calculate L = lim(n→∞) |aₙ₊₁/aₙ|. If L < 1: absolutely convergent. If L > 1 or L = ∞: divergent. If L = 1: inconclusive.

p-Series Test - When to use

Use for series of form Σ1/nᵖ. Converges if p > 1. Diverges if p ≤ 1. Examples: Σ1/n² converges (p=2), Σ1/√n diverges (p=1/2).

Comparison Test - When to use

Use when you can compare to a known series (geometric, p-series, etc.). If 0 ≤ aₙ ≤ bₙ and Σbₙ converges, then Σaₙ converges. If aₙ ≥ bₙ ≥ 0 and Σbₙ diverges, then Σaₙ diverges.

Limit Comparison Test - When to use

Use when direct comparison is hard but series "behave similarly". Calculate lim(n→∞) aₙ/bₙ = L. If 0 < L < ∞, then Σaₙ and Σbₙ either both converge or both diverge.

Integral Test - When to use

Use when aₙ = f(n) where f is positive, continuous, and decreasing. Σaₙ and ∫f(x)dx from 1 to ∞ either both converge or both diverge.

Geometric Series - When to use

For series of form Σarⁿ. Converges if |r| < 1 (sum = a/(1-r)). Diverges if |r| ≥ 1.

Strategy: See factorial or exponential

Use RATIO TEST. Factorials and exponentials make the ratio simplify nicely.

Strategy: See alternating signs

First try ALTERNATING SERIES TEST. If that shows convergence, check ABSOLUTE CONVERGENCE by testing Σ|aₙ| to determine if absolutely or conditionally convergent.

Strategy: See 1/nᵖ form

Use P-SERIES TEST directly. Remember: converges if p > 1, diverges if p ≤ 1.

Strategy: Series looks like a simpler known series

Use COMPARISON TEST or LIMIT COMPARISON TEST. Compare to geometric, p-series, or other known series.

Strategy: Can easily integrate the function

Use INTEGRAL TEST if the function is positive, continuous, and decreasing.

Quick Check: Before any other test

ALWAYS check TEST FOR DIVERGENCE first! If lim(n→∞) aₙ ≠ 0, you're done - it diverges.

Factorial in numerator vs denominator

Factorial in NUMERATOR (n!/100ⁿ): Usually diverges - factorial grows faster than exponential. Factorial in DENOMINATOR (10ⁿ/n!): Usually converges - factorial in denominator dominates.

When Ratio Test gives L = 1

The test is INCONCLUSIVE. Try a different test (comparison, integral, etc.). Don't give up!

Absolute vs Conditional Convergence - Testing

To check absolute convergence: test Σ|aₙ|. If it converges → absolutely convergent. If it diverges BUT original series converges → conditionally convergent.