Ratio Test

Ratio Test: Let $\sum a_n$ be a real or complex series with $a_n\neq 0$ eventually. Define $\rho=\limsup_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|.$

• If $\rho<1$, then $\sum a_n$ converges absolutely .
• If $\rho>1$ (or $\rho=\infty$), then $\sum a_n$ diverges .
• If $\rho=1$, the test is inconclusive.

The ratio test is particularly effective for factorials and exponential-type terms.

Proof sketch (optional): If $\rho<1$, choose $r$ with $\rho<r<1$ so that eventually $|a_{n+1}|\le r|a_n|$, implying $|a_n|$ is bounded by a geometric sequence and hence summable.