Question

Evaluate the given infinite geometric series described as 3 + 2 + 4/3 + 8/9 ...

Answer 1

Given infinite geometric series is

`3+2+(4)/(3)+(8)/(9)+.\ldots.`

The first term of the series is 3 and the common ratio is

`(2)/(3)<1`

For a finite geometric series with first term a and common ratio r, the sum is

`S=(a(1-r^n))/(1-r)`

Now, as n tends to infinity,

`r^n\rightarrow0`

as when r<1

For the given problem

`r=(2)/(3)<1`

Therefore, the sum is

`\begin{gathered} S=(a)/(1-r) \\ =(3)/(1-(2)/(3)) \\ =(3)/((1)/(3)) \\ =9 \end{gathered}`