Question

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

Answer 1

Given infinite geometric series is

$3+2+\frac{4}{3}+\frac{8}{9}+.\ldots.$

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

$\frac{2}{3}<1$

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

$S=\frac{a(1-r^n)}{1-r}$

Now, as n tends to infinity,

$r^n\rightarrow0$

as when r<1

For the given problem

$r=\frac{2}{3}<1$

Therefore, the sum is

$\begin{gathered} S=\frac{a}{1-r} \\ =\frac{3}{1-\frac{2}{3}} \\ =\frac{3}{\frac{1}{3}} \\ =9 \end{gathered}$