Q. 57 - Calculus

Question

Use the divergence test to prove that a geometric series $\sum _{k=1}^{\infty } c{r}^k$diverges when $\left|r\right| \ge 1$ and $c \ne 0.$

Step-by-Step Solution

Answer

$$\begin{gathered} \mathrm{The} \mathrm{series} \mathrm{does} \mathrm{not} \mathrm{converge} \mathrm{to} 0. \\ \mathrm{Therefore}, \mathrm{the} \mathrm{series} \mathrm{is} \mathrm{divergent} \mathrm{by} \mathrm{divergence} \mathrm{test}. \\ \mathrm{Thus}, \mathrm{by} \mathrm{divergence} \mathrm{test}, \mathrm{the} \mathrm{geometric} \mathrm{series} \sum _{\mathrm{k}=1}^{\infty } {\mathrm{cr}}^{\mathrm{k}} \mathrm{diverges} \mathrm{when} \left|\mathrm{r}\right|\ge 1 \mathrm{and} \mathrm{c}\ne 0. \end{gathered}$$

Step 1. Given information is:

$$\begin{gathered} \sum _{k=1}^{\infty } c{r}^k and \\ \left|r\right|\ge 1 , c\ne 0 \end{gathered}$$

Step 2. Examining different cases for c>0

$$\begin{gathered} \mathrm{First}, \mathrm{assume} \mathrm{c} > 0 \\ \mathrm{Case} 1: \mathrm{when} \mathrm{the} \mathrm{ratio} \mathrm{r}\le -1 \\ \mathrm{The} \mathrm{ratio} \mathrm{of} \mathrm{geometric} \mathrm{series} \sum _{\mathrm{k}=1}^{\infty } {\mathrm{cr}}^{\mathrm{k}} \mathrm{is} \mathrm{r}\le -1, \\ \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \underset{\mathrm{k}\to \infty }{\lim } {\mathrm{cr}}^{\mathrm{k}} \mathrm{does} \mathrm{not} \mathrm{exist}. \\ \mathrm{Case} 2: \mathrm{When} \mathrm{the} \mathrm{ratio} \mathrm{r}=1 \\ \mathrm{The} \mathrm{ratio} \mathrm{of} \mathrm{geometric} \mathrm{series} \sum _{\mathrm{k}=1}^{\infty } {\mathrm{cr}}^{\mathrm{k}} \mathrm{is} \mathrm{r}=1, \\ \mathrm{then} \mathrm{the} \mathrm{series} \mathrm{become} \sum _{\mathrm{k}=1}^{\infty } {\mathrm{cr}}^{\mathrm{k}} = \sum _{\mathrm{k}=1}^{\infty } \mathrm{c} \\ \mathrm{Therefore}, \underset{\mathrm{k}\to \infty }{\lim } {\mathrm{cr}}^{\mathrm{k}} = \mathrm{c} \\ \mathrm{The} \mathrm{sequence} \mathrm{does} \mathrm{not} \mathrm{converge} \mathrm{to} 0. \\ \mathrm{Therefore}, \mathrm{for} \mathrm{r}=1, \mathrm{the} \mathrm{series} \mathrm{is} \mathrm{divergent} \mathrm{by} \mathrm{divergence} \mathrm{test}. \\ \mathrm{Case} 3: \mathrm{When} \mathrm{the} \mathrm{ratio} \mathrm{r}>1 \\ \mathrm{The} \mathrm{value} \mathrm{of} \underset{\mathrm{k}\to \infty }{\lim } {\mathrm{cr}}^{\mathrm{k}} \mathrm{is}: \\ \underset{\mathrm{k}\to \infty }{\lim } {\mathrm{cr}}^{\mathrm{k}} = \infty \\ \mathrm{Therefore}, \mathrm{the} \mathrm{sequence} \mathrm{does} \mathrm{not} \mathrm{converge} \mathrm{to} 0. \end{gathered}$$

Step 3. Result

$$\begin{gathered} \mathrm{Therefore}, \mathrm{the} \mathrm{series} \mathrm{is} \mathrm{divergent} \mathrm{by} \mathrm{divergence} \mathrm{test}. \\ \mathrm{In} \mathrm{the} \mathrm{similar} \mathrm{fashion}, \mathrm{result} \mathrm{is} \mathrm{proved} \mathrm{for} \mathrm{c}<0. \\ \mathrm{Thus}, \mathrm{by} \mathrm{divergence} \mathrm{test}, \mathrm{the} \mathrm{geometric} \mathrm{series} \sum _{\mathrm{k}=1}^{\infty } {\mathrm{cr}}^{\mathrm{k}} \mathrm{diverges} \mathrm{when} \left|\mathrm{r}\right|\ge 1 \mathrm{and} \mathrm{c}\ne 0. \end{gathered}$$