Question 1092410
The ratio of each term to the prior is 1/3. 
Sequence is 1/3, 1/9, 1/27, 1/81, 1/243.  As n increases, lim of 1/3^n =0.  That helps but doesn't prove convergence.
partial sums are (108/243) and 36/243, and 4/243. This will converge to 0.
The ratio a/1-r is 1/3/1-(1/3)=(1/3)/(2/3)=(1/2).
The series itself converges to 0, and the sum of the series, if one starts at 1/3, converges to (1/2).  If it started with 1, then it would converge to (3/2).