Nth Root Test/Weak Form

Theorem

Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be a series of (strictly) positive real numbers $\R$.

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

Then:

If $l > 1$, the series $\ds \sum_{n \mathop = 1}^\infty a_n$ diverges.

If $l < 1$, the series $\ds \sum_{n \mathop = 1}^\infty a_n$ converges absolutely.

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Cauchy convergence test
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Cauchy convergence test

• krm2233 (https://math.stackexchange.com/users/1155637/krm2233), Definition of Cauchy's Convergence Test using $\lim$ rather than $\limsup$, URL (version: 2023-10-15): https://math.stackexchange.com/q/4787186