Determine whether the following series converges or diverges: sum_(k = 1)

Ernstfalld 2021-08-12 Answered
Determine whether the following series converges or diverges:
\(\displaystyle{\sum_{{{k}={1}}}^{\infty}}\frac{{3}}{{\left({k}+{4}\right)}^{{3}}}\)

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Expert Answer

Velsenw
Answered 2021-08-13 Author has 23795 answers

\(\displaystyle{\sum_{{{k}={1}}}^{\infty}}\frac{{3}}{{\left({k}+{4}\right)}^{{3}}}=\)
\(\displaystyle={\sum_{{{k}={1}}}^{\infty}}\frac{{3}}{{k}^{{3}}}\)
By p-series test \(\displaystyle{\sum_{{{k}={1}}}^{\infty}}\frac{{1}}{{h}^{{p}}},{p}{>}{1}\) – converges, \(\displaystyle{p}{<}{1}\) – diverges.
So this series converges, cause it’s a p-series with p = 3.

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