\(\displaystyle{\sum_{{{k}={1}}}^{\infty}}\frac{{{\left(-{1}\right)}^{k}\times{k}}}{{{6}{k}^{4}+{1}}}\)

\(\displaystyle{a}_{{k}}=\frac{k}{{{6}{k}^{4}+{1}}}\)

\(\displaystyle{b}_{{k}}=\frac{k}{{{6}{k}^{4}+{1}}}\)

1) \(\displaystyle{b}_{{k}}\ge{0}\)

2) \(\displaystyle\lim_{{{k}+{0}}}\frac{k}{{{6}{k}^{4}+{1}}}={0}\)

Then, by alternative series test we can converge.

The term of series are alternative and limit of absolute value is 0, so series converge by alternative series test.