Chapter 8.3, Problem 82E

To determine

The sum of the given infinite geometric series.

Answer to Problem 82E

The sum is divergent.

Explanation of Solution

Given information: The infinite geometric series $2+\frac{7}{3}+\frac{49}{18}+\frac{343}{108}+\mathrm{...}$

Concept used:

In a sequence $a_1,a_2,a_3,a_4,\dots \dots ,a_n,\dots$ if ratio of the consecutive terms is the same then it is defined as Geometric Sequence.

So, a number r is defined as the common ratio of the geometric sequence if $\frac{a_2}{a_1}=\frac{a_3}{a_2}=\frac{a_4}{a_3}=\dots \dots =r.$

The sum of an infinite geometric series $a_1+a_{1}r+a_1{r}^2+a_1{r}^3+\cdots +a_1{r}^{n-1}+\cdots$ is $S={\displaystyle {\sum }_{i=0}^{\infty }{a}_1{r}^i=\frac{a_1}{1-r}}$ where $\left|r\right|<1$ .

Calculation:

Consider the series $2+\frac{7}{3}+\frac{49}{18}+\frac{343}{108}+\mathrm{...}$

Here the first term $t_1=a=2$ , second term $t_2=\frac{7}{3}$

Then the common ratio $r=\frac{t_2}{t_1}=\frac{\frac{7}{3}}{2}=\frac{7}{6}$

As, $\left|r\right|=\frac{7}{6}>1$

So, the sum is divergent.

It is not possible to find out the sum of the infinite series.