Nth Root Test/Warning
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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.
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 6.18$