Chapter 11.3, Problem 41E

(a)

To determine

To prove: That $\left(1-r\right)\left(a+ar+a{r}^2+\cdot \cdot \cdot +a{r}^n\right)=a-a{r}^{n+1}$ and from this conclude that, for

$r\ne 1$, $a+ar+a{r}^2+\cdot \cdot \cdot +a{r}^n=\frac{a}{1-r}-\frac{a{r}^{n+1}}{1-r}$.

(b)

To determine

Why the geometric series ${\displaystyle {\sum }_0^{\infty }a{r}^k}$ converges to $\frac{a}{1-r}$ when $\left|r\right|<1$.

(c)

To determine

Why the geometric series ${\displaystyle {\sum }_0^{\infty }a{r}^k}$ diverges for $\left|r\right|>1$.

(d)

To determine

Why the geometric series ${\displaystyle {\sum }_0^{\infty }a{r}^k}$ diverges for $r=1$ and $r=-1$.