Q. 16 - Calculus

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 $\sum _{k=1}^{\infty }{b}_k$.

Step-by-Step Solution

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 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

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