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Determine the convergence of the series (-1)n / n1/n

Consider the series

    \[ \sum_{n=1}^{\infty} \frac{(-1)^n}{n^{\frac{1}{n}}}. \]

Determine whether the series converges or diverges. If it converges, determine whether it converges conditionally or absolutely.


The given series diverges.

Proof. The series diverges since the limit

    \[ \lim_{n \to \infty} \frac{(-1)^n}{n^{\frac{1}{n}}} \neq 0. \]

(Since the limit of the even terms is 1 and the limit of the odd terms is -1.) \qquad \blacksquare

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