Ratio test

College/University

Definition

A convergence test for a series when . It states that a series converges absolutely if L < 1. A series is divergent if L > 1. One of these two results must be met or the limit does not exist. If L = 1 the series may diverge or converge thus rendering the test inconclusive.

Worked examples

\(\sum_{n=1}^{\infty} \frac{2^n}{n!}\): \(L = \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n\to\infty} \frac{2^{n+1}}{(n+1)!} \cdot \frac{n!}{2^n} = \lim_{n\to\infty} \frac{2}{n+1} = 0\)

Since \(L = 0 < 1\), the series converges absolutely by the ratio test.

\(\sum_{n=1}^{\infty} \frac{n^3}{3^n}\): \(L = \lim_{n\to\infty} \left|\frac{(n+1)^3}{3^{n+1}} \cdot \frac{3^n}{n^3}\right| = \lim_{n\to\infty} \frac{(n+1)^3}{3n^3} = \frac{1}{3}\)

Since \(L = \frac{1}{3} < 1\), the series converges absolutely.

\(\sum_{n=1}^{\infty} n!\): \(L = \lim_{n\to\infty} \frac{(n+1)!}{n!} = \lim_{n\to\infty} (n+1) = \infty\)

Since \(L = \infty > 1\), the series diverges.

Common mistakes

\(L = 1\) so the series diverges → \(L = 1\) means the ratio test is inconclusive When \(L = 1\), the series may converge or diverge; you must use a different test.
\(\lim_{n\to\infty} \frac{a_n}{a_{n+1}}\) (ratio flipped) → \(\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|\) (next over current) The ratio test requires the next term divided by the current term, not the reverse.
Forgetting absolute value: \(\lim_{n\to\infty} \frac{a_{n+1}}{a_n}\) → \(\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|\) The ratio test requires the absolute value of the ratio to handle alternating series correctly.

Where you'll use it next

You'll use the ratio test throughout calculus when analyzing power series, Taylor series, and determining radius of convergence. It appears in differential equations and advanced topics where infinite series model functions.

Found in 1 StudyPug lesson

Mastering the Ratio Test for Series Convergence

Calculus 2

Unlock the power of the ratio test to analyze series convergence. Learn step-by-step techniques, understand key conditions, and gain confidence in solving complex calculus problems.

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