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

Question
Functions
asked 2021-02-13
Does \(\sum−1^{n}\ln 2n^{\frac{1}{n}})\) converge or diverge?

Answers (1)

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.
0

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