Question

Determine whether the series converges: sum_(k = 1)^oo 4/(k + 2)^5

Series
ANSWERED
asked 2021-09-10
Determine whether the series converges:
\(\displaystyle{\sum_{{{k}={1}}}^{\infty}}\frac{{4}}{{\left({k}+{2}\right)}^{{5}}}\)

Expert Answers (1)

2021-09-11

\(\displaystyle{\sum_{{{k}={1}}}^{\infty}}\frac{{4}}{{\left({k}+{2}\right)}^{{5}}}\)
\(\displaystyle{a}_{{k}}=\frac{{4}}{{\left({k}+{2}\right)}^{{5}}}\)
\(\displaystyle{a}_{{k}}=\frac{{4}}{{{k}^{{5}}{\left({1}+\frac{{2}}{{k}}\right)}^{{5}}}}\)
Let assume that,
\(\displaystyle{b}_{{k}}=\frac{{1}}{{k}^{{5}}}\)
By the limit comparison test,
\(\displaystyle{L}=\lim_{{{k}\to\infty}}\frac{{a}_{{k}}}{{b}_{{k}}}\)
\(\displaystyle{L}=\lim_{{{k}\to\infty}}\frac{{\frac{{4}}{{{k}^{{5}}{\left({1}+\frac{{2}}{{k}}\right)}^{{5}}}}}}{{\frac{{1}}{{k}^{{5}}}}}\)
\(\displaystyle{L}=\lim_{{{k}\to\infty}}\frac{{4}}{{\left({1}+\frac{{2}}{{k}}\right)}^{{5}}}\)
\(\displaystyle{L}=\frac{{4}}{{\left({1}+\frac{{2}}{\infty}\right)}^{{5}}}\)
\(\displaystyle{L}=\frac{{4}}{{\left({1}+{0}\right)}^{{5}}}\)
\(\displaystyle{L}={4}\)
Here \(\displaystyle{L}={4}\) – finite and non-zero, therefore the series is a p-series with \(\displaystyle{p}={5}{\left({b}_{{k}}=\frac{{1}}{{k}^{{5}}}\right)}\), so the given series is converges by the properties of p-series.
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