Answer
Converges, sum: $\frac{9}{2}$
Work Step by Step
An infinite geometric series converges if and only if $|r|\lt1$, where $r$ is the common ratio. If it converges, then it equals $\frac{a_1}{1-r}$ where $a_1$ is the first term.
The common ratio is the quotient of two consecutive terms: $r=\frac{a_2}{a_1}=\dfrac{1}{3}=\frac{1}{3}$, $|\frac{1}{3}|\lt1$, thus it converges.
Hence the sum (since $a_1=1$): $\dfrac{3}{1-\frac{1}{3}}=\frac{9}{2}$
