Convergence tests
Objective. Select comparison, ratio, root and alternating tests.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Convergence tests are systematic criteria used to decide whether an infinite series converges when direct summation is impossible. The comparison test compares a series of positive terms to a known convergent or divergent one; the ratio test examines L = lim |aₙ₊₁/aₙ|, concluding convergence if L < 1 and divergence if L > 1; the root test uses lim |aₙ|^(1/n) similarly; and the alternating series test confirms convergence for Σ(−1)ⁿbₙ when bₙ decreases to 0. Choosing the right test depends on the series' structure: factorials suggest the ratio test, nth powers suggest the root test, and alternating signs suggest the alternating test. For example, Σ nⁿ/n! has ratio aₙ₊₁/aₙ → e, which exceeds 1, so it diverges.
Worked example
Use a convergence test to determine whether the series Σ (from n=1 to ∞) of n/2ⁿ converges.
- 1Apply the ratio test: compute aₙ₊₁/aₙ = [(n+1)/2ⁿ⁺¹] ÷ [n/2ⁿ].
- 2Simplify the ratio to (n+1)/(2n).
- 3Take the limit as n → ∞: (n+1)/(2n) → 1/2.
- 4Since 1/2 < 1, the series converges by the ratio test.
Answer: Converges, since the ratio test limit is 1/2 < 1
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Start practisingCommon mistake. A frequent error is applying the ratio test and concluding convergence when the limit equals exactly 1; this case is inconclusive, as Σ 1/n diverges while Σ 1/n² converges, both giving ratio limit 1.
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