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

Graphical or geometric meaning

Eventually, each term is roughly half the previous one, so the tail fits beneath a scaled geometric decay. A log-scale plot of term magnitudes becomes approximately linear when the ratio stabilizes below one.

Common mistakes and why they fail

Verification and reasonableness checks

• Estimate consecutive numerical ratios for large $n$.
• Ensure divergence conclusion agrees with the term test when $L>1$.
• Use a known $p$-series to remember why $L=1$ is inconclusive.

Interpret all three outcomes

Let $L=\lim|a_{n+1}/a_n|$ when it exists. The series converges absolutely if $L<1$, diverges if $L>1$ or is infinite, and remains unclassified by this test if $L=1$. Both the harmonic series and a convergent $p=2$ series give the boundary result, so it cannot be forced into a conclusion. Factorials and exponentials simplify well in consecutive-term ratios. Cancel before taking the limit and retain absolute values to measure magnitude. For a power series, solve the inequality in $|x-c|$, then test endpoints separately because the ratio often becomes one there. Always state when the test is inconclusive rather than calling that outcome divergence.

Practice

1. Test $\sum1/n!$.
2. What does the ratio test say about $\sum1/n$?
3. Test $\sum n!/3^n$.

Connections and next steps

• Root Test
• Radius of Convergence
• Absolute and Conditional Convergence