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An infinite geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. One of the magical aspects of mathematics is that under certain conditions, even though we are adding up an infinite number of terms, the series has a finite sum.

For a series to be convergent, and thus for us to find a finite sum, the common ratio must have an absolute value less than 1. This ensures that the terms decrease in magnitude and get closer and closer to zero as we move further along in the series.

In the context of the given exercise, we've confronted an infinite geometric series in the form of \(\sum_{n=0}^{\infty} 2\left(-\frac{2}{3}\right)^{n}\). Initially, this might seem confusing, but we can analyze and grasp the concept by knowing that each term is expressed as the product of an initial term and a common ratio raised to an increasing power.

The notion of a common ratio is pivotal in understanding geometric series. It is the constant factor between consecutive terms of the series. If you have any two successive terms, the common ratio (\(r\)) can be found by dividing one term by the one that comes before it.

For example, to find the common ratio in our given series, we compare the term when \(n = 1\) to the term when \(n = 0\). This yields \(\frac{2\left(-\frac{2}{3}\right)^{1}}{2\left(-\frac{2}{3}\right)^{0}} = -\frac{2}{3}\), confirming that \(r = -\frac{2}{3}\). Having a common ratio whose absolute value is less than 1 means our series converges, and in our example, the absolute value of \(\frac{2}{3}\) is indeed less than 1, leading to a sum that can be adequately found.

The series summation formula provides us with a straightforward method to find the sum of an infinite geometric series. This formula is denoted as \(S = \frac{a}{1 - r}\), where \(S\) is the sum of the series, \(a\) is the first term, and \(r\) is the common ratio. When |r| < 1, the sum converges to a finite value.

By substituting the first term (2) and the common ratio (-2/3) into the formula, the sum of the series can be determined as shown in our exercise solution above. Simplification is a crucial step in finding the final value of the sum. Here, we inverted the denominator and then multiplied it by the numerator to get the final result, \(\frac{6}{5}\) or 1.2. It's through using this neat formula that we can sum an infinite number of terms simply and elegantly.