Discuss the convergence or divergence of the series .
Let us compare this series with the geometric series .
If r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge. Since in our case, r is equal to 1/3, the geometric series must converge.
Thus, since this limit is finite and positive, both series converge or diverge. Since the geometric series converges, then our series must also converge.