Decision guideCalculus IIIntermediate10 min

Ratio test or root test? Choose from the series structure

Use ratios for factorial and product growth, roots for whole expressions raised to n, and recognize when either test returns no decision.

LaTeX article Updated July 13, 2026

L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|\quad\text{or}\quad L=\lim_{n\to\infty}\sqrt[n]{|a_n|}

Ratios cancel factorial and product structure

The ratio aₙ₊₁/aₙ is efficient when shifting n to n + 1 creates large cancellations, especially with factorials, exponentials, and consecutive products.

Write the shifted term carefully before dividing. Most ratio-test errors come from an incorrect (n + 1)! or exponent.

\frac{(n+1)!}{n!}=n+1

Nth roots expose repeated powers

The root test is natural when the entire term has the form [bₙ]ⁿ or contains several factors raised to n. The nth root removes that outer exponent immediately.

Absolute values are built into both tests. A conclusion L < 1 therefore proves absolute convergence, which is stronger than conditional convergence.

\sqrt[n]{\left|\left(\frac{2n}{3n+1}\right)^n\right|}=\left|\frac{2n}{3n+1}\right|

A limit of one means change methods

When L = 1, neither test says the series converges or diverges. Harmonic and p-series examples show why no universal conclusion is possible.

Move to comparison, integral, alternating-series, or another structure-appropriate test rather than repeating the same inconclusive calculation.

\sum\frac1n:\ L=1\text{ and diverges}\qquad\sum\frac1{n^2}:\ L=1\text{ and converges}

Worked example

Common mistakes

Declaring convergence when L = 1.
Forgetting absolute values around a sign-changing term.
Using the ratio test when an nth root would remove the main complexity at once.

Keep these ideas

Factorials favor ratios.
Whole nth powers favor roots.
L = 1 requires a different test.