Your ratio-test limit is 1. What next?

The three outcomes of the ratio test

For a series with nonzero terms eventually, compute L = lim |aₙ₊₁ / aₙ| as n grows. If the limit exists, these are the useful cases:

• L < 1: the series converges absolutely.
• L > 1, including an infinite limit: the series diverges.
• L = 1: the test is inconclusive; try a different argument.

Two limits of 1, two different results

For aₙ = 1/n, the ratio is n/(n + 1), which approaches 1. The harmonic series diverges. For aₙ = 1/n², the ratio is n²/(n + 1)², which also approaches 1. That series converges by the p-series test. The same ratio-test limit can sit behind opposite outcomes.

Inspect the form before choosing again

Ask what the original series resembles. Powers of n may suggest a p-series or comparison. Alternating signs may suggest the alternating-series test. A suitable positive decreasing function may suggest an integral test. Check the conditions of whichever test you choose.

• First check whether the terms approach 0. If they do not, the series diverges.
• Terms approaching 0 alone do not establish convergence.
• Keep the original series visible while you choose the next test.

A factorial example with a different outcome

For aₙ = n!/3ⁿ, cancellation gives aₙ₊₁ / aₙ = (n + 1)/3. This grows without bound, so the ratio test does establish divergence. The useful step is the simplification before you take the limit.

Explain your next step

Try writing one sentence before doing more algebra: “The ratio-test limit is 1, so I will use ___ because the original terms have ___.” A reasoned choice is more useful than repeating a test that has already told you it cannot decide.