Root Criterion

analysis

Definition

Root Criterion

The root criterion is a convergence criterion for infinite series.

1. $\exists q<1:\sqrt[n]{|a_n|}\le q$ for almost all $n$ $⟹{\sum }_{n{\ge }_0}{a}_n$ absolute convergent
2. $\sqrt[n]{|a_n|}\ge 1$ for infinite number of $n$ $⟹{\sum }_{n\ge 0}{a}_n$ divergent

Limit (superior) form:

3. $\lim {\sup }_{n\to \infty }\sqrt[n]{|a_n|}<1⟹{\sum }_{n\ge 0}{a}_n$ convergent
4. $\lim {\sup }_{n\to \infty }\sqrt[n]{|a_n|}>1⟹{\sum }_{n\ge 0}{a}_n$ divergent

Clearly visible in the limit form, the root criterion does not apply where $\sqrt[n]{|a_n|}=1$.