Use the ratio test to find whether the following series converge or diverge: $$\sum_{n=0}^{\infty} \frac{3^{2 n}}{2^{3 n}}$$

When studying infinite series, one of the key concepts is 'convergence'. A series converges if the sum of its terms approaches a specific finite number as the number of terms goes to infinity. To explain it simply, imagine trying to add up an endless list of numbers. If those numbers get smaller quickly enough, their total won't keep growing indefinitely. Instead, it'll level off near a particular value. For example, the series \(\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + ...\) converges to 1. This means that as you keep adding more and more terms of this series, the total gets closer and closer to 1, but never surpasses it. The ratio test helps us determine if a series converges by examining the ratio of successive terms in the series.

Next, we have 'divergence', which is the opposite of convergence. A series diverges if the sum of its terms does not approach a specific finite value as the number of terms goes to infinity. In simpler words, if you keep adding more terms to the series and the total keeps growing without bounds, or wiggles endlessly without settling to a number, it diverges. For example, the series \(\frac{3^2}{2^3} + \frac{3^{4}}{2^{6}} + \frac{3^{6}}{2^{9}} + ...\) diverges because, as we found using the ratio test, the ratio of the successive terms is \(\frac{9}{8}\), which is greater than 1. This result tells us that each new term added makes the total keep increasing without bound, hence the series does not settle to any finite value.

An 'infinite series' is simply an expression that represents adding an infinite sequence of terms. Consider it as trying to add an endless list. Some famous examples include geometric series and harmonic series. These infinite series can either converge to a number or diverge, based on the behavior of their terms. The concept of infinite series is crucial in various branches of mathematics and science as it helps model phenomena or solve problems involving continuous processes. The ratio test is a handy tool for determining the behavior of these series. It works by evaluating the limit of the ratio of successive terms. If the ratio turns out to be less than 1, the series converges. If more than 1, the series diverges. If equal to 1, the test is inconclusive, and further tests are needed.