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Convergence of sequences and series
Lesson 6 of 10
Standard practice

Tests for series convergence

Objective. Apply the ratio, comparison and nth-term tests.

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  2. 2. Worked example
  3. 3. Practice
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Learn: 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.

1/2
Fraction bar showing 1 out of 2 parts shaded
  1. 1Ratio of consecutive terms: (1/2ⁿ⁺¹)/(1/2ⁿ) = 1/2.
  2. 2Limit is 1/2 < 1.

Answer: Converges

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Common mistake. Concluding convergence just because the terms tend to zero.

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