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Understanding whether an infinite series converges or diverges is a fundamental aspect when dealing with sequences and series. A series converges if its terms approach a specific value as you add more and more terms; otherwise, it diverges.

An infinite geometric series, which consists of terms in a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio, will converge if the absolute value of the common ratio is less than 1. If the absolute value is 1 or greater, the series will not settle towards a specific value and therefore diverges.

For the exercise given, by checking the common ratios, \(\frac{2}{5}\) and \(\frac{5}{7}\), since they are both less than 1 in absolute value, we can confirm that each individual geometric series converges. This is an essential first step to establish before we attempt to find the sum of the series.

For a geometric series that converges, you can calculate its sum using a straightforward formula: \[S = \frac{a}{1-r}\] Here, \(S\) is the sum of the series, \(a\) is the first term of the series, and \(r\) is the common ratio. Essentially, what this formula does is it accounts for all the terms in the series by considering the limiting behavior as the number of terms goes to infinity.

From the exercise, once we establish convergence, applying the formula provides us with the sums for each series separately. By substituting the relevant values into the formula, \(S1\) for the first series using \(a=3\) and \(r=\frac{2}{5}\), and \(S2\) for the second series with \(a=2\) and \(r=\frac{5}{7}\), it yields the finite sums for the otherwise infinitely long series.

The common ratio in a geometric series is what determines the series' behavior. It is the factor by which any term of the series is multiplied to get the next term. In the case of the exercise provided, the two given series have common ratios of \(\frac{2}{5}\) and \(\frac{5}{7}\) respectively.

These values are vital because they not only dictate convergence but also influence the value of the sum if the series converges. Geometric series with a common ratio between -1 and 1 (except for 0) will always converge to a sum, whereas those outside of this interval will not. Knowing this, we see why determining the common ratio is a step that cannot be skipped in analyzing geometric series. For the exercise series, both common ratios meet this criterion for convergence, thus assuring us of a meaningful sum as seen in the previous steps.