Nth Root Test/Warning

Nth Root Test: Warning
Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be a series of real numbers $\R$ or complex numbers $\C$.

Let the sequence $\sequence {a_n}$ be such that the limit superior $\ds \limsup_{n \mathop \to \infty} \size {a_n}^{1/n} = l$.

If $l = 1$, the Nth Root Test provides no information on whether $\ds \sum_{n \mathop = 1}^\infty a_n$ converges absolutely, converges conditionally, or diverges.

If $\ds \limsup_{n \mathop \to \infty} \size {a_n}^{1/n} = \infty$, then of course $\ds \sum_{n \mathop = 1}^\infty a_n$ diverges.