Series Convergence Tests Overview

Overview

This lecture reviews key tests to determine whether an infinite series converges or diverges, including their conditions and examples of applying each.

Divergence Test

• Take the limit of aₙ as n approaches infinity; if the limit ≠ 0, the series diverges.
• If the limit = 0, the test is inconclusive; use another test.

Geometric Series Test

• Series has 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) for |r| < 1.

p-Series Test

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

Telescoping Series

• Terms cancel in sequence, leaving only first and last terms of partial sum.
• Write the partial sum formula, then take the limit as n→∞.
• If the limit is finite, the series converges.

Integral Test

• Define f(x) where aₙ = f(n); f(x) must be positive, continuous, and decreasing for large x.
• If ∫f(x)dx from 1 to ∞ is finite, series converges; if not, it diverges.

Ratio Test

• Compute limₙ→∞ |aₙ₊₁ / aₙ|.
• If the result < 1, the series converges; if > 1, it diverges; if = 1, test is inconclusive.

Root Test

• Compute limₙ→∞ ⁿ√|aₙ|.
• Result < 1 means convergence; > 1 means divergence; = 1 is inconclusive.

Direct Comparison Test

• Compare a series aₙ with a known bₙ.
• If bₙ ≥ aₙ ≥ 0 and ∑bₙ converges, then ∑aₙ also converges.
• If aₙ ≥ bₙ ≥ 0 and ∑bₙ diverges, then ∑aₙ diverges.

Limit Comparison Test

• Take limₙ→∞ aₙ/bₙ = L, where 0 < L < ∞.
• If ∑bₙ converges, so does ∑aₙ, and vice versa.

Alternating Series Test

• Applies to series with alternating signs (e.g., (−1)ⁿaₙ).
• If limₙ→∞ aₙ = 0 and aₙ is decreasing, series converges.

Absolute and Conditional Convergence

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

Key Terms & Definitions

• Converge — The sum of the infinite series approaches a finite value.
• Diverge — The sum of the series does not approach a finite value.
• Geometric Series — Series with a constant ratio between terms.
• p-Series — Series of the form 1/nᵖ.
• Telescoping Series — Series where intermediate terms cancel.
• Integral Test — Uses an improper integral to test for convergence.
• Ratio/Root Test — Tests involving limits of term ratios or roots.
• Direct/Limit Comparison — Comparing with known series.
• Alternating Series — Series with terms that alternate in sign.
• Absolute Convergence — Convergence when all terms are positive.
• Conditional Convergence — Series converges only with alternating signs.

Action Items / Next Steps

• Practice identifying which test to apply to a given series.
• Work through textbook problems using each test.
• Review any series not easily classified by the above tests.