Chapter 12.3, Problem 4AYU

To determine

To fill: The value of the infinite geometric series ${\displaystyle {\sum }_{k=1}^{\infty }a{r}^{k-1}}$ if the common ratio $\left|r\right|<1$.

Answer to Problem 4AYU

Solution:

The sum of the geometric series ${\displaystyle {\sum }_{k=1}^{\infty }a{r}^{k-1}=\frac{a_1}{1-r}}$ if $\left|r\right|<1$.

Explanation of Solution

Given:

Common ratio $\left|r\right|<1$ and the infinite geometric series is ${\displaystyle {\sum }_{k=1}^{\infty }a{r}^{k-1}}$.

Calculation:

The Convergence of an infinite Geometric series theorem states that if $\left|r\right|<1$, the infinite geometric series ${\displaystyle {\sum }_{k=1}^{\infty }a{r}^{k-1}}$ converges. Its sum is ${\displaystyle {\sum }_{k=1}^{\infty }a{r}^{k-1}=\frac{a_1}{1-r}}$.