Series Convergence Tests Overview

Overview

This lecture reviews the main convergence and divergence tests for infinite series, explaining how and when to use each test with examples.

Divergence Test

• Take the limit as n→∞ of aₙ; if the limit ≠ 0, the series diverges.
• If the limit = 0, the test is inconclusive; try another test.

Geometric Series Test

• Series of the form a·rⁿ; identify the common ratio r.
• If |r| < 1, the series converges; if |r| ≥ 1, it diverges.
• The sum is a₁/(1 – r) if the series converges.

P-Series Test

• Series of the form 1/nᵖ.
• If p > 1, the series converges; if p ≤ 1, it diverges.

Telescoping Series

• Write terms to reveal cancellation between positive and negative fractions.
• Find the partial sum formula, then take its limit as n→∞.
• If the limit is finite, the series converges.

Integral Test

• Replace aₙ with a function f(x) that is positive, continuous, and decreasing for x ≥ N.
• Integrate f(x) from N to ∞; if the result is finite, the series converges; otherwise, it diverges.

Ratio Test

• Compute limit as n→∞ of |aₙ₊₁ / aₙ|.
• If the limit < 1, series converges; if > 1, diverges; if = 1, inconclusive.

Root Test

• Compute limit as n→∞ of ⁿ√|aₙ|.
• If the limit < 1, series converges; if > 1, diverges; if = 1, inconclusive.

Direct Comparison Test

• Compare a series aₙ to a known series bₙ.
• If bₙ ≥ aₙ ≥ 0 and bₙ converges, then aₙ converges.
• If 0 ≤ bₙ ≤ aₙ and bₙ diverges, then aₙ diverges.

Limit Comparison Test

• Take limit as n→∞ of aₙ / bₙ = L, where 0 < L < ∞.
• If known series bₙ converges (or diverges), aₙ converges (or diverges) as well.

Alternating Series Test

• Series of the form (–1)ⁿaₙ.
• Requires: (1) limₙ→∞ aₙ = 0, and (2) aₙ is decreasing.
• If both are met, series converges.

Absolute and Conditional Convergence

• If ∑|aₙ| converges, the series is absolutely convergent.
• If ∑aₙ converges but ∑|aₙ| diverges, the series is conditionally convergent.

Key Terms & Definitions

• Converges — Series approaches a finite sum.
• Diverges — Series does not approach a finite sum.
• Geometric Series — Series with constant ratio between terms.
• P-Series — Series of the form 1/nᵖ.
• Telescoping Series — Series where terms cancel each other in sequence.
• Integral Test — Uses improper integrals to test convergence.
• Ratio Test — Uses ratio of successive terms.
• Root Test — Uses nth root of terms.
• Direct Comparison — Compares terms to a known series.
• Limit Comparison — Compares limit of ratio of terms to a known series.
• Alternating Series — Series with terms alternating in sign.
• Absolute/Conditional Convergence — Distinction based on convergence of ∑|aₙ|.

Action Items / Next Steps

• Practice identifying which test to apply based on series structure.
• Review example problems for each convergence/divergence test.
• Complete assigned exercises applying these tests to new series.