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

Zoe Oneal 2021-08-18 Answered

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

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Expert Answer

Ayesha Gomez
Answered 2021-08-19 Author has 11266 answers

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

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