# Classify whether this series convergessum_(k = 1)^oo ((-1)^k xx k)

Classify whether this series converges $$\displaystyle{\sum_{{{k}={1}}}^{\infty}}\frac{{{\left(-{1}\right)}^{k}\times{k}}}{{{6}{k}^{4}+{1}}}$$

• Questions are typically answered in as fast as 30 minutes

### Plainmath recommends

• Get a detailed answer even on the hardest topics.
• Ask an expert for a step-by-step guidance to learn to do it yourself.

Ayesha Gomez

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