Short answer: run through the tests in a fixed order. Start with the divergence test (if the terms don't go to 0, the series diverges). Then check whether it is a geometric or p-series. Then use the ratio test for factorials and exponentials, comparison for fractions of powers, and the alternating series test when signs alternate.
The checklist
- Divergence test: if lim an ≠ 0, the series diverges. If the limit is 0, this test tells you nothing.
- Geometric series Σ arn: converges if |r| < 1, to a/(1 − r).
- p-series Σ 1/np: converges if p > 1, diverges if p ≤ 1. The harmonic series (p = 1) diverges.
- Factorials or n-th powers (n!, 3n, nn): use the ratio test. L = lim |an+1/an|. L < 1 converges, L > 1 diverges, and L = 1 is inconclusive.
- Fractions of polynomials or roots: use the limit comparison test against the p-series made from the highest powers.
- (−1)n in the terms: use the alternating series test. If |an| decreases to 0, the series converges.
- A function you can integrate easily: use the integral test (positive, decreasing, continuous terms).
Worked example: Σ n / 2n
There is an exponential, so use the ratio test: |an+1/an| = (n + 1)/(2n) → 1/2 < 1. Converges.
Worked example: Σ (n + 1) / (n³ + 2)
The highest powers give n/n³ = 1/n². Limit comparison with Σ 1/n²: the ratio of terms → 1, a finite positive number, and Σ 1/n² converges (p = 2). Converges.
Mistakes that cost marks
- Concluding "converges" because the terms go to 0. The harmonic series is the counterexample.
- Using the ratio test on p-series, where it always gives L = 1 and decides nothing.
- Skipping the conditions. Comparison and integral tests require positive terms.
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