91Ó°ÊÓ

Consider the following given information.

(a) a factorial, where ratio test can determine whether series is convergence or divergence.

(b) a factorial series on which the ratio test will be ineffective in determining convergence or divergence.

(c) A series on which the root test will be effective in determining convergence or divergence.

Consider a series $\underset{k=0}{\overset{∞}{∑}}{a}_k=\underset{k=0}{\overset{∞}{∑}}\frac{5^k}{k!}$and determine the value of $\mathrm{ÒÏ}=\underset{k→∞}{\lim }\frac{a_{k+1}}{a_k}.$

localid="1661334327520" $\begin{array}{rcl}\mathrm{ÒÏ}& =& \underset{k→∞}{\lim }\frac{a_{k+1}}{a_k}\\ & =& \underset{k→∞}{\lim }\frac{\frac{5^{k+1}}{(k+1)!}}{\frac{5^k}{k!}}\\ & =& \underset{k→∞}{\lim }\frac{5^k·5^1·k!}{5^k·k!·(k+1)}\\ & =& \underset{k→∞}{\lim }\frac{5}{k+1}\\ & =& 0\end{array}$

Here $\mathrm{ÒÏ}<1,$so the series converges according to the ratio test.

Consider a series $\underset{k=1}{\overset{∞}{∑}}{a}_k=\underset{k=1}{\overset{∞}{∑}}\frac{2!}{k^2}$and determine the value of $\mathrm{ÒÏ}=\underset{k→∞}{\lim }\frac{a_{k+1}}{a_k}.$

$\begin{array}{rcl}\mathrm{ÒÏ}& =& \underset{k→∞}{\lim }\frac{a_{k+1}}{a_k}\\ & =& \underset{k→∞}{\lim }\frac{\frac{2!}{{(k+1)}^2}}{\frac{2!}{k^2}}\\ & =& \underset{k→∞}{\lim }\frac{2!·k^2}{2!·{(k+1)}^2}\\ & =& \underset{k→∞}{\lim }\frac{k^2}{{(k+1)}^2}\\ & =& 1\end{array}$

Here $\mathrm{ÒÏ}=1$so the ratio test is inconclusive.

Consider a series $\underset{k=1}{\overset{∞}{∑}}{a}_k=\underset{k=1}{\overset{∞}{∑}}\frac{3^k}{k^5}$and determine the value of $\mathrm{ÒÏ}=\underset{k→∞}{\lim }{\left(a_k\right)}^{\frac{1}{k}}$.

$\begin{array}{rcl}\mathrm{ÒÏ}& =& \underset{k→∞}{\lim }{\left(a_k\right)}^{\frac{1}{k}}\\ & =& \underset{k→∞}{\lim }{\left(\frac{3^k}{k^5}\right)}^{\frac{1}{k}}\\ & =& \underset{k→∞}{\lim }\left(\frac{3^{{\displaystyle \frac{k}{k}}}}{k^{{\displaystyle \frac{5}{k}}}}\right)\\ & =& 3\end{array}$

Here $\mathrm{ÒÏ}>1$so the series converges according to the root test.