Convergence and divergence of series — A/L Combined Mathematics

How do I practise Convergence and divergence of series?

1 question with a worked solution

Check before you answer: Partial Sum Closed Form: Finding the exact algebraic expression for S_n before taking the limit as n → ∞.

1. Under what condition on the common ratio r does an infinite geometric series a + ar + ar² + ... (with a ≠ 0) converge to a finite sum?
   1. A) r = 1
   2. B) |r| < 1 (-1 < r < 1)
   3. C) r > 1
   4. D) |r| ≤ 1
   Check my answer

   Correct answer: B) |r| < 1 (-1 < r < 1)

   The n-th partial sum of a geometric series is S_n = a(1 - rⁿ) / (1 - r). As n → ∞, rⁿ → 0 if and only if |r| < 1. In this case, the series converges to the finite sum S_∞ = a / (1 - r). If |r| ≥ 1, the series diverges.

   Show worked solution

   Step 1: S_n = a(1 - rⁿ) / (1 - r). Step 2: For convergence, lim_n→∞ rⁿ = 0, which requires |r| < 1. Answer: |r| < 1 (-1 < r < 1).
   Modelled on: the A/L Combined Mathematics examination pattern

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Where does Convergence and divergence of series fit in the syllabus?

Year 1 · Term 3