Question

If $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=\infty$and$\sum _{k=1}^{\infty }{a}_k$ diverges, explain why we cannot draw any conclusions about the behavior of$\sum _{k=1}^{\infty }{b}_k$.

Short answer

$$\begin{gathered} \mathrm{The}\mathrm{behavior}\mathrm{of}\mathrm{the}\mathrm{series}\underset{\mathrm{k}=1}{\overset{\infty }{\sum {\mathrm{b}}_{\mathrm{k}}}}\mathrm{cannot}\mathrm{be}\mathrm{determined}\mathrm{when}\underset{\mathrm{k}\to \infty }{\lim }\frac{{\mathrm{a}}_{\mathrm{k}}}{{\mathrm{b}}_{\mathrm{k}}}=\infty \\ \mathrm{and}\underset{\mathrm{k}=1}{\overset{\infty }{\sum {\mathrm{a}}_{\mathrm{k}}}}\mathrm{is}\mathrm{divergent}\mathrm{holds}\mathrm{because}\mathrm{the}\mathrm{series}\underset{\mathrm{k}=1}{\overset{\infty }{\sum {\mathrm{a}}_{\mathrm{k}}}}\mathrm{may}\mathrm{or}\mathrm{may}\mathrm{not} \\ \mathrm{be}\mathrm{divergent},\mathrm{even}\mathrm{though}\underset{\mathrm{k}\to \infty }{\lim }\frac{{\mathrm{a}}_{\mathrm{k}}}{{\mathrm{b}}_{\mathrm{k}}}=\infty \mathrm{holds}. \end{gathered}$$

Step-by-step solution

Step 1. Given information

$\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=\infty and\sum _{k=1}^{\infty }{a}_{k}di\mathrm{verges}$

Step 2. Finding the value of limit 

Consider the convergent series $\sum _{k=1}^{\infty }{a}_k=\sum _{k=1}^{\infty }\frac{1}{k}$and the series $\sum _{k=1}^{\infty }{b}_k=\frac{1}{k^2}$.

$$\begin{gathered} \mathrm{The}\mathrm{value}\mathrm{of}\underset{\mathrm{k}\to \infty }{\lim }\frac{{\mathrm{a}}_{\mathrm{k}}}{{\mathrm{b}}_{\mathrm{k}}}\mathrm{is}: \\ \underset{\mathrm{k}\to \infty }{\lim }\frac{{\mathrm{a}}_{\mathrm{k}}}{{\mathrm{b}}_{\mathrm{k}}}=\underset{\mathrm{k}\to \infty }{\lim }\frac{{\displaystyle \frac{1}{\mathrm{k}}}}{{\displaystyle \frac{1}{{\mathrm{k}}^2}}} \\ =\underset{\mathrm{k}\to \infty }{\lim }\mathrm{k} \\ =\infty \end{gathered}$$

Step 3. Result

$$\begin{gathered} \mathrm{The}\mathrm{series}\underset{\mathrm{k}=1}{\overset{\infty }{\sum {\mathrm{b}}_{\mathrm{k}}}}=\frac{1}{{\mathrm{k}}^2}\mathrm{is}\mathrm{convergent}\mathrm{but}\mathrm{the}\mathrm{series}\underset{\mathrm{k}=1}{\overset{\infty }{\sum {\mathrm{a}}_{\mathrm{k}}}}=\frac{1}{\mathrm{k}}\mathrm{is}\mathrm{divergent}. \\ \mathrm{The}\mathrm{behavior}\mathrm{of}\mathrm{the}\mathrm{series}\underset{\mathrm{k}=1}{\overset{\infty }{\sum {\mathrm{b}}_{\mathrm{k}}}}\mathrm{cannot}\mathrm{be}\mathrm{determined}\mathrm{when}\underset{\mathrm{k}\to \infty }{\lim }\frac{{\mathrm{a}}_{\mathrm{k}}}{{\mathrm{b}}_{\mathrm{k}}}=\infty \\ \mathrm{and}\underset{\mathrm{k}=1}{\overset{\infty }{\sum {\mathrm{a}}_{\mathrm{k}}}}\mathrm{is}\mathrm{divergent}\mathrm{holds}\mathrm{because}\mathrm{the}\mathrm{series}\underset{\mathrm{k}=1}{\overset{\infty }{\sum {\mathrm{a}}_{\mathrm{k}}}}\mathrm{may}\mathrm{or}\mathrm{may}\mathrm{not}\mathrm{be} \\ \mathrm{divergent},\mathrm{even}\mathrm{though}\underset{\mathrm{k}\to \infty }{\lim }\frac{{\mathrm{a}}_{\mathrm{k}}}{{\mathrm{b}}_{\mathrm{k}}}=\infty \mathrm{holds}. \end{gathered}$$