10.8 Ratio Test for Convergence

This whole unit is BC only. Let L = lim (n → ∞) |aₙ₊₁/aₙ|. Then:

• L < 1: Σ aₙ converges absolutely (so it converges).
• L > 1 (including L = ∞): Σ aₙ diverges.
• L = 1: the test is inconclusive. Use a different test.

If the ratio of consecutive terms approaches L < 1, then for large n the series behaves like a geometric series with ratio about L, which converges. If L > 1, the terms eventually grow, so they can't approach 0. The ratio test is really a comparison with a geometric series.

Write aₙ₊₁ by replacing every n with n + 1, then divide by aₙ by flipping and multiplying. These facts do most of the work:

• (n + 1)! = (n + 1)·n!, so (n + 1)!/n! = n + 1.
• (2n + 2)! = (2n + 2)(2n + 1)·(2n)!.
• aⁿ⁺¹/aⁿ = a.
• (n + 1)ᵏ/nᵏ → 1 as n → ∞, for any fixed power k.

• Terms don't go to 0? nth term test: diverges.
• Geometric (n in the exponent, constant ratio)? Use |r| < 1.
• Looks like 1/nᵖ? p-series test.
• Rational function or roots of n? Compare (direct or limit) with a p-series.
• Alternating signs? Alternating series test, and check absolute values for absolute convergence.
• Factorials or n in an exponent? Ratio test.
• Easy antiderivative and positive, decreasing terms? Integral test.

Reach for the ratio test when the terms have factorials or n in an exponent, or both, as in n²/3ⁿ or 5ⁿ/n!. For terms that are rational functions of n, like 1/n² or n/(n³ + 1), the ratio always tends to 1, so the test is useless. Use p-series or comparison tests for those.

The ratio test is also the main tool for finding the radius of convergence of a power series (10.13).

You won't be tested on other convergence tests, such as the root test. The tests on the exam are the nth term, integral, direct comparison, limit comparison, alternating series and ratio tests, plus geometric and p-series.