Unit 10: Infinite Sequences & Series

Series Convergence Tests

Applying the integral, comparison, ratio, and alternating series tests.

Advanced25 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Sequences & Convergence

From Finite Sums to Infinite Ones

An infinite series is defined as the limit of its partial sums: Sₙ = a₁+a₂+...+aₙ, and the series converges to limₙ→∞ Sₙ if that limit exists. Before reading on: for a geometric sum a+ar+ar²+...+ar^(n−1), could multiplying the whole sum by r, then subtracting, cancel almost everything and leave a clean formula?

Worked Example — Deriving the Geometric Series Formula

Let Sₙ = a+ar+ar²+...+ar^(n−1). Then rSₙ = ar+ar²+...+arⁿ. Subtracting: Sₙ−rSₙ = a−arⁿ, so Sₙ(1−r) = a(1−rⁿ), giving Sₙ = a(1−rⁿ)/(1−r) for r ≠ 1. If |r| < 1, rⁿ → 0 as n → ∞, so Sₙ → a/(1−r) — the infinite geometric series sum, derived from the finite formula's limit. Apply it: Σₙ₌₀^∞ 3(1/4)ⁿ has a = 3, r = 1/4, converging to 3/(1−1/4) = 3/(3/4) = 4.

\sum_{n=0}^{\infty} a r^n = \frac{a}{1-r}, \quad |r| < 1

Worked Example — The Ratio Test

Determine whether Σₙ₌₁^∞ n/2ⁿ converges. The ratio test examines L = limₙ→∞ |aₙ₊₁/aₙ|: [(n+1)/2^(n+1)] / [n/2ⁿ] = [(n+1)/n]·(1/2) → (1)(1/2) = 1/2 as n → ∞. Since L = 1/2 < 1, the series converges.

Worked Example — The Alternating Series Test

Determine whether Σₙ₌₁^∞ (−1)^(n+1)/n converges. The terms 1/n are positive, decreasing, and → 0 as n → ∞ — all three conditions of the alternating series test are met, so the series converges. This is notable: the non-alternating harmonic series, Σ1/n, actually diverges — alternating its signs is exactly what makes the difference.

Tip

The ratio test is inconclusive exactly at L = 1 — a different test is needed in that case, since the series could converge or diverge.

Common Mistakes

Assuming a series converges just because its terms go to 0.
Terms going to 0 is necessary for convergence but not sufficient — the harmonic series Σ1/n has terms going to 0 and still diverges.

Key Takeaways

An infinite series' value is defined as the limit of its partial sums.
The geometric series formula a/(1−r) (for |r|<1) is derived from the finite partial-sum formula's limit.
The ratio test and alternating series test each determine convergence from the terms' own behavior, without needing an exact sum.

Summary

The next lesson generalizes a series to include a variable term, x, and asks: for which values of x does it actually converge?

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