Question

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

Series
Determine whether the series converges:
$$\displaystyle{\sum_{{{k}={1}}}^{\infty}}\frac{{4}}{{\left({k}+{2}\right)}^{{5}}}$$

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.