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.