AP Calc BC: Which Series Convergence Test to Use (decision tree)

AP Calc BC tests 8 series convergence tests and most students try them in random order. The decision tree below picks the right test in under 10 seconds for any series the exam can throw at you.

The decision tree

  1. Geometric series? (form Σ arn) → converges iff |r| < 1. Done.
  2. p-series? (form Σ 1/np) → converges iff p > 1. Done.
  3. Telescoping? (terms cancel like 1/n − 1/(n+1)) → partial sum test. Done.
  4. Term-by-term comparison possible? (close to a known geometric or p-series) → Comparison Test.
  5. Terms look like a ratio with factorials or exponentials? → Ratio Test.
  6. Terms look like an n-th power? → Root Test.
  7. Alternating signs? → Alternating Series Test.
  8. None of the above → try Integral Test if an = f(n) for a nice f, or Limit Comparison Test.

Quick examples

Σ n!/nn → ratio of factorials and powers → Ratio Test. Limit is 1/e < 1, so converges.

Σ (3n/4n) → geometric with r = 3/4 < 1, converges.

Σ 1/(n² + 1) → close to p-series 1/n² (p=2 > 1), use Comparison Test, converges.

Σ (−1)n/n → alternating series with terms decreasing to 0, Alternating Series Test, converges (conditionally).

Σ 1/(n ln n) → not p-series, not geometric, use Integral Test on f(x) = 1/(x ln x). Integral diverges, so series diverges.

Mistakes that cost AP exam points

  1. Forgetting to check that an → 0. If the terms don't approach zero, the series diverges. This is the simplest check and the easiest point.
  2. Using the Ratio Test on a p-series. The ratio test limit is 1 for any p-series, giving you no information. Use the p-series test directly.
  3. Applying the Alternating Series Test without checking decreasing terms. The terms must be decreasing in absolute value, not just alternating in sign.
  4. Confusing absolute and conditional convergence. Σ (−1)n/n converges by AST but Σ 1/n diverges, so Σ (−1)n/n is conditionally convergent. The exam asks about this distinction.

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