Sequences and Series — Convergence Tests

GRE Math Subject · Axiom Academy

LESSON Sequences and Series — Convergence Tests Critical tests for determining series behavior Divergence Test (nth-Term Test) If lim(n→∞) aₙ ≠ 0, the series Σ aₙ DIVERGES. Quick check: If individual terms don't go to 0, the series diverges for sure. Example: Σ 1/n² converges (p=2>1), Σ 1/n diverges (p=1) • If L = 1: INCONCLUSIVE (use other tests) Best for: Factorials, exponentials Same conclusions as ratio test. Useful for aₙ = (bₙ)ⁿ terms. • If Σ bₙ converges, then Σ aₙ converges • If Σ aₙ diverges, then Σ bₙ diverges If f is positive, decreasing, continuous: Σ f(n) and ∫₁^∞ f(x) dx both converge or both diverge Converges if: (1) lim aₙ = 0 and (2) aₙ is decreasing ✓ Series convergence is a high-frequency topic on the Subject GRE. Memorize these tests and practice applying them to unfamiliar series.

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