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

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

Question
Functions
asked 2021-03-18
Does \(\displaystyle\sum−{1}^{{{n}}}{\ln{{2}}}{n}^{{{\frac{{{1}}}{{{n}}}}}}{)}\) converge or diverge?

Answers (1)

2021-03-19

Let \(\displaystyle{a}_{{{n}}}={\left(-{1}\right)}^{{{n}}}{\log{{\left({2}{n}^{{{\frac{{{1}}}{{{2}}}}}}\right)}}}.\) Here
\(\lim_{n \rightarrow \infty}|an| = \lim_{n \rightarrow \infty} \log(2n^{\frac{1}{n}})| =\log(\lim 2n^{\frac{1}{n}}) (log is continuous function in [1,∞) ) =\log2 (Since \lim_{n \rightarrow \infty} n^{\frac{1}{n}}=1\)\(\displaystyle{\sum_{{{n}={1}}}^{\infty}}{\left(-{1}\right)}^{{{n}}}{\log{{\left({2}{n}^{{{\frac{{{1}}}{{{n}}}}}}\right)}}}\) is not convergent.

 
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