Answer
This series converges.
$S = 2$
Work Step by Step
To determine whether or not this series converges or diverges, we need to find the common ratio, $r$. This can be done by setting up a ratio of the second term to the first term:
$r = \frac{\frac{4}{9}}{\frac{2}{3}}$
Simplify by multiplying the numerator by the reciprocal of the denominator:
$r = \frac{4}{9} \times \frac{3}{2}$
Multiply:
$r = \frac{12}{18}$
Simplify:
$r = \frac{2}{3}$
When $|r| \lt 1$, then the series converges; when $|r| \gt 1$, then the series diverges. Since $|r| \lt 1$ in this case, this series converges.
We can now use the following formula to calculate the sum of a convergent geometric series:
$S = \frac{a_1}{1 - r}$
Plug in our known values:
$S = \frac{\frac{2}{3}}{1 - \frac{2}{3}}$
Subtract first:
$S = \frac{\frac{2}{3}}{\frac{1}{3}}$
Divide fractions by multiplying the numerator by the reciprocal of the denominator:
$S = \frac{2}{3} \times \frac{3}{1}$
Multiply:
$S = \frac{6}{3}$
Simplify:
$S = 2$
