1. The problem asks whether the infinite series $$\sum_{k=1}^\infty \frac{5}{3^{k-1}}$$ is convergent or divergent. 2. This is a geometric series with the first term $$a = \frac{5}{3^{0}} = 5$$ and common ratio $$r = \frac{1}{3}$$ because each term is multiplied by $$\frac{1}{3}$$ to get the next term. 3. A geometric series $$\sum_{k=0}^\infty ar^k$$ converges if and only if $$|r| < 1$$. If it converges, the sum is $$\frac{a}{1-r}$$. 4. Here, $$|r| = \frac{1}{3} < 1$$, so the series converges. 5. The sum of the series is $$\frac{5}{1 - \frac{1}{3}} = \frac{5}{\frac{2}{3}} = 5 \times \frac{3}{2} = \frac{15}{2} = 7.5$$. 6. Therefore, the series is convergent and its sum is $$7.5$$.