Ratio Test

The central idea

For a series $\sum a_n$, let $L=\lim|a_{n+1}/a_n|$ when the limit exists. If $L<1$, the series converges absolutely; if $L>1$ or $L=\infty$, it diverges. If $L=1$, the ratio test is inconclusive. The test is especially effective for factorials and exponential powers.

Definitions, hypotheses, and notation

The limit form can be weakened to limsup for a more general theorem, but the familiar three-case rule covers most structured examples. If $L<1$, choose a number $r$ with $L<r<1$; eventually $|a_{n+1}|\le r|a_n|$, producing a geometric bound on the tail. This is the proof mechanism.

Polynomial factors do not affect an exponential limiting ratio because $(n+1)^k/n^k\to1$. Factorials do affect it strongly because $(n+1)!/n!=n+1$. Recognizing those cancellations makes the test both fast and reliable.

Conceptual meaning

The ratio compares late terms with a geometric series. A limiting shrink factor below one gives geometric-like decay; a factor above one prevents terms from tending to zero. A factor tending to one is too finely balanced for the test to decide.

A dependable method and decision rule

1. Write $a_{n+1}$ carefully with every $n$ shifted.
2. Form the absolute ratio and cancel factorial or power factors.
3. Take the limit.
4. Compare the result strictly with one.
5. If the result equals one, select another test rather than forcing a conclusion.

Fully worked example

Common mistakes and why they fail