Root Test

Root Test: For a series $\sum a_n$ (real or complex), define $\alpha=\limsup_{n\to\infty}\sqrt[n]{|a_n|}.$

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

The root test is well-suited for expressions like $|a_n|=(\text{something})^n$.

Proof sketch (optional): If $\alpha<1$, choose $r$ with $\alpha<r<1$. Then for large $n$, $|a_n|\le r^n$, and $\sum r^n$ converges.