Choosing a Convergence Test

Every convergence test is built to notice one kind of structure. Ratio and root tests are built for terms with factorials, n-th powers, or things raised to the n. Comparison tests (direct or limit comparison) are built for terms that look like a simpler series once you squint - a rational function of n behaves like a power of 1/n for large n. The integral test is built for terms that come from a positive, decreasing, continuous function you can actually integrate. The alternating series test is built for terms with a clean (-1)ⁿ or (-1)ⁿ⁺¹ sign flip and otherwise decreasing size. Reading the term's shape tells you which machine was built to open that lock.

Work (n+1)/(n²+3n): for large n this looks like n/n² = 1/n, so limit comparison with the divergent harmonic series 1/n gives a finite positive limit and the series diverges. The ratio test would give a limit of 1 here - no information - because the terms are already a rational function of n, not an exponential or factorial one. That silence is the giveaway: ratio and root tests only decide series whose terms grow or shrink geometrically or factorially. For sum of (-1)ⁿ n/2ⁿ, the sign-flip plus the 2ⁿ in the denominator points to two options: check absolute convergence with the ratio test, since factor n/2ⁿ has an n-th power, and the ratio of consecutive terms tends to 1/2, so it converges absolutely.

This is the same judgment call from choosing an integration technique in Unit 6: you scanned an integrand for a product (parts), a composition (substitution), or a rational function (partial fractions) before picking a method. Here you scan a term for a power, a factorial, a rational expression, or a sign change before picking a test. Both skills reward pattern recognition over brute force, and both get faster once you stop testing every method and start reading the expression first.