\(\displaystyle{\sum_{{{k}={1}}}^{\infty}}{\left(\frac{{2}}{{\left({k}+{5}\right)}^{{3}}}\right)}\)

\(\displaystyle\lim_{{{k}\to\infty}}{\left({a}_{{k}}\right)}=\lim_{{{\left({k}\to{5}\right)}^{{5}}}}{\left(\frac{{2}}{{\left({k}+{5}\right)}^{{3}}}\right)}=\frac{{9}}{\infty}={0}\)

The limit of the terms of the series is 0, so the series diverges by the divergent test.

\(\displaystyle\lim_{{{k}\to\infty}}{\left({a}_{{k}}\right)}=\lim_{{{\left({k}\to{5}\right)}^{{5}}}}{\left(\frac{{2}}{{\left({k}+{5}\right)}^{{3}}}\right)}=\frac{{9}}{\infty}={0}\)

The limit of the terms of the series is 0, so the series diverges by the divergent test.