Question

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

Short answer

$$\begin{gathered} \mathrm{The}\mathrm{behaviour}\mathrm{of}\mathrm{the}\mathrm{series}\sum _{\mathrm{k}=1}^{\infty }{\mathrm{b}}_{\mathrm{k}}\mathrm{cannot}\mathrm{be}\mathrm{determined}\mathrm{when}\underset{\mathrm{k}\to \infty }{\lim }\frac{{\mathrm{a}}_{\mathrm{k}}}{{\mathrm{b}}_{\mathrm{k}}}=0\mathrm{and}\sum _{\mathrm{k}=1}^{\infty }{\mathrm{a}}_{\mathrm{k}}\mathrm{is}\mathrm{convergent}\mathrm{holds} \\ \mathrm{because}\mathrm{the}\mathrm{series}\sum _{\mathrm{k}=1}^{\infty }{\mathrm{b}}_{\mathrm{k}}\mathrm{may}\mathrm{or}\mathrm{may}\mathrm{be}\mathrm{not}\mathrm{convergent},\mathrm{even}\mathrm{though}\underset{\mathrm{k}\to \infty }{\lim }\frac{{\mathrm{a}}_{\mathrm{k}}}{{\mathrm{b}}_{\mathrm{k}}}=0\mathrm{holds}. \end{gathered}$$

Step-by-step solution

Step 1. Given information

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

Step 2. Explanation

$$\begin{gathered} \sum _{k=1}^{\infty }{a}_k=\frac{1}{k^2}\mathrm{and}\mathrm{the}\mathrm{series}\sum _{k=1}^{\infty }{b}_k=\sum _{k=1}^{\infty }\frac{1}{k} \\ \mathrm{The}\mathrm{value}\mathrm{of}\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=\underset{k\to \infty }{\lim }\frac{\frac{1}{k^2}}{\frac{1}{k}} \\ \underset{k\to \infty }{\lim }\frac{1}{k} \\ =0 \end{gathered}$$

Step 3. Result 

$$\begin{gathered} \sum _{k=1}^{\infty }{a}_k=\frac{1}{k^2}i\mathrm{s}\mathrm{divergen}t\mathrm{and}\mathrm{the}\mathrm{series}\sum _{k=1}^{\infty }{b}_k=\sum _{k=1}^{\infty }\frac{1}{k}\mathrm{is}\mathrm{convergent} \\ \mathrm{The}\mathrm{behaviour}\mathrm{of}\mathrm{the}\mathrm{series}\sum _{\mathrm{k}=1}^{\infty }{\mathrm{b}}_{\mathrm{k}}\mathrm{cannot}\mathrm{be}\mathrm{determined}\mathrm{when}\underset{\mathrm{k}\to \infty }{\lim }\frac{{\mathrm{a}}_{\mathrm{k}}}{{\mathrm{b}}_{\mathrm{k}}}=0\mathrm{and}\sum _{\mathrm{k}=1}^{\infty }{\mathrm{a}}_{\mathrm{k}}\mathrm{is}\mathrm{convergent}\mathrm{holds} \\ \mathrm{because}\mathrm{the}\mathrm{series}\sum _{\mathrm{k}=1}^{\infty }{\mathrm{b}}_{\mathrm{k}}\mathrm{may}\mathrm{or}\mathrm{may}\mathrm{be}\mathrm{not}\mathrm{convergent},\mathrm{even}\mathrm{though}\underset{\mathrm{k}\to \infty }{\lim }\frac{{\mathrm{a}}_{\mathrm{k}}}{{\mathrm{b}}_{\mathrm{k}}}=0\mathrm{holds}. \end{gathered}$$