Decision guideCalculus IIIntermediate13 min read

How to choose a convergence test

Match the series’ structure to a test instead of cycling through tests in chapter order.

\sum a_n\quad\longrightarrow\quad\text{structure first}

Decision guide

A practical comparison that turns a vague choice into a repeatable test.

Reviewed July 11, 2026

DecisionStart here

Start with the nth-term check, then identify geometric, p-series, telescoping, alternating, factorial/exponential, or power-series structure before choosing a comparison, ratio, root, or integral test.

Run the fast checks

If a_n does not approach zero, stop: the series diverges. Next look for an exact geometric ratio, a p-series, or cancellation in partial sums.

These direct recognitions produce stronger and shorter conclusions than a general-purpose test.

Compare positive-term series by dominant behavior

Rational expressions in n behave like the ratio of their leading powers. Factorials and exponentials usually favor the Ratio Test. Expressions raised to the nth power often favor the Root Test.

Choose a benchmark with known behavior and a meaningful asymptotic relationship.

Conditional versus absolute convergence

For alternating series, first test absolute values. Absolute convergence settles the question strongly. If the absolute series diverges, the Alternating Series Test may still establish conditional convergence when magnitudes decrease to zero.

Worked exampleChoose by dominant powers

Test Σ (3n + 1)/(n³ + 2).

\frac{3n+1}{n^3+2}\sim\frac{3n}{n^3}=\frac3{n^2}

Identify the dominant behavior.

\sum\frac1{n^2}\ \text{converges}

Use the p-series benchmark p = 2.

\lim_{n\to\infty}\frac{(3n+1)/(n^3+2)}{1/n^2}=3

A positive finite limit validates limit comparison.

Result

\boxed{\text{The series converges by limit comparison.}}

Watch for

Common mistakes

Skipping the nth-term divergence check.
Using the Ratio Test on rational powers where its limit is inconclusive.
Claiming an alternating series converges without checking decreasing magnitudes and zero limit.

Keep

Three takeaways

Recognize special forms before general tests.
Dominant behavior suggests the comparison target.
Distinguish absolute from conditional convergence.