# Does sum−1^{n}ln 2n^{frac{1}{n}}) converge or diverge?

Question
Functions
Does $$\sum−1^{n}\ln 2n^{\frac{1}{n}})$$ converge or diverge?

2021-02-14
Let $$a_{n}= (-1)^{n}\log(2n^{\frac{1}{2}}).$$ Here
$$\lim_{n \rightarrow \infty}|an| = \lim_{n \rightarrow \infty} \log(2n^{\frac{1}{n}})| =\log(\lim 2n^{\frac{1}{n}}) (log is continuous function in [1,∞) ) =\log2 (Since \lim_{n \rightarrow \infty} n^{\frac{1}{n}}=1$$
$$\sum_{n=1}^\infty (-1)^{n}\log(2n^{\frac{1}{n}})$$ is not convergent.

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