Convergence of sequences and series
Lesson 6 of 10
Tests for series convergence
Objective. Apply the ratio, comparison and nth-term tests.
The key idea
If terms do not tend to zero the series diverges. Otherwise compare with a known series, or use the ratio test: L < 1 converges, L > 1 diverges, L = 1 is inconclusive.
Worked example
Test Σ 1/2ⁿ using the ratio test.
- 1Ratio of consecutive terms: (1/2ⁿ⁺¹)/(1/2ⁿ) = 1/2.
- 2Limit is 1/2 < 1.
Answer: Converges
Practice
Each question comes with a picture and a listen button. Try it first, then reveal the answer.
Common mistake. Concluding convergence just because the terms tend to zero.
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