Q. 17 - Calculus

Question

Consider the series 

 Fill in the blanks and select the correct word: 

$If \underset{k\to \infty }{\lim }\frac{a_k}{b_k}=\infty \mathrm{and} \sum _{k=1}^{\infty } \_\_\_\_\_ \mathrm{diverges} t\mathrm{he}n \sum _{k=1}^{\infty } \_\_\_\_\mathrm{\_} (\mathrm{converges}/\mathrm{diverges}).$

Step-by-Step Solution

Answer

$\mathrm{The} \mathrm{first} \mathrm{blank} \mathrm{is} \mathrm{filled} \mathrm{with} \sum _{\mathrm{k}=1}^{\infty } {\mathrm{b}}_{\mathrm{k} } \mathrm{and} \mathrm{second} \mathrm{with} \sum _{\mathrm{k}=1}^{\infty } {\mathrm{a}}_{\mathrm{k}}.$

Step 1. Given information

$\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=\infty$.

Step 2. Limit comparison test

$$\begin{gathered} \mathrm{The} \mathrm{limit} \mathrm{comparison} \mathrm{test} \mathrm{states} \mathrm{that} \mathrm{for} \mathrm{and} \mathrm{be} \mathrm{two} \mathrm{series} \mathrm{with} \mathrm{positive} \mathrm{terms} \mathrm{then} \\ \mathrm{If} \underset{\mathrm{k}\to \infty }{\lim }\frac{{\mathrm{a}}_{\mathrm{k}}}{{\mathrm{b}}_{\mathrm{k}}}=\mathrm{L} \mathrm{Where} \mathrm{L} \mathrm{is} \mathrm{any} \mathrm{positive} \mathrm{real} \mathrm{number} \mathrm{then} \mathrm{either} \mathrm{both} \mathrm{converges} \mathrm{or} \mathrm{diverges}. \\ \mathrm{If} \underset{\mathrm{k}\to \infty }{\lim }\frac{{\mathrm{a}}_{\mathrm{k}}}{{\mathrm{b}}_{\mathrm{k}}}=0 \mathrm{and} \sum _{\mathrm{k}=1}^{\infty } {\mathrm{b}}_{\mathrm{k} }\mathrm{converges} \mathrm{then} \sum _{\mathrm{k}=1}^{\infty } {\mathrm{a}}_{\mathrm{k}} \mathrm{converges}. \\ \mathrm{If} \underset{\mathrm{k}\to \infty }{\lim }\frac{{\mathrm{a}}_{\mathrm{k}}}{{\mathrm{b}}_{\mathrm{k}}}=\infty \mathrm{and} \sum _{\mathrm{k}=1}^{\infty } {\mathrm{b}}_{\mathrm{k} } \mathrm{diverges} \mathrm{then} \sum _{\mathrm{k}=1}^{\infty } {\mathrm{a}}_{\mathrm{k}} \mathrm{diverges}. \end{gathered}$$

Step 3. Result

$$\begin{gathered} \mathrm{If} \underset{\mathrm{k}\to \infty }{\lim }\frac{{\mathrm{a}}_{\mathrm{k}}}{{\mathrm{b}}_{\mathrm{k}}}=\infty \mathrm{and} \sum _{\mathrm{k}=1}^{\infty } {\mathrm{b}}_{\mathrm{k} } \mathrm{diverges} \mathrm{then} \sum _{\mathrm{k}=1}^{\infty } {\mathrm{a}}_{\mathrm{k}} \mathrm{diverges}. \\ \mathrm{Therefore} , \mathrm{the} \mathrm{first} \mathrm{blank} \mathrm{is} \mathrm{filled} \mathrm{with} \sum _{\mathrm{k}=1}^{\infty } {\mathrm{b}}_{\mathrm{k} } \mathrm{and} \mathrm{second} \mathrm{with} \sum _{\mathrm{k}=1}^{\infty } {\mathrm{a}}_{\mathrm{k}}. \end{gathered}$$