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A geometric series is a sequence of numbers where each term after the first is found by multiplying the preceding term by a fixed, non-zero number called the common ratio. For instance, in the series 2, 6, 18, 54, ..., the common ratio is 3 because each term is three times larger than the term before it.

Geometric series can take on various forms, but they are widely used in fields such as finance for calculating compounded interest, in physics for waves and optics, and in computer science for algorithm analysis. The general formula for the nth term of a geometric series is given by
\[ a_n = a_1 \times r^{(n-1)} \]
where \(a_1\) is the first term and \(r\) is the common ratio. To find the sum of a finite geometric series up to n terms, the formula is
\[ S_n = \frac{a_1(1 - r^n)}{1 - r} \]
provided that \(r\) is not equal to 1. This formula helps summing a geometric series efficiently without needing to add all terms individually.

The convergence of a series refers to the idea that as more and more terms are added together, the series approaches a certain value, known as the sum of the series. Not all series converge; if a series does not approach a finite value, it is said to diverge.

An infinite geometric series will converge if its common ratio \(r\) is between -1 and 1, i.e., \(|r| < 1\). The sum to which a convergent series approaches can be found using the sum formula for an infinite geometric series. For example, a series with a common ratio of 0.5 will converge, but if the common ratio were 1.5, the series would diverge, with its terms increasing without bound.

In our exercise, the common ratio is 0.2, which is less than 1, so the series converges. This fundamental property of convergence is used to determine whether it's possible to calculate a definitive sum for an infinite geometric series using the formula \( S = \frac{a}{1 - r} \).

An infinite series is the sum of the terms of an infinite sequence. While it might seem counterintuitive that you can assign a finite number to the sum of infinitely many terms, in mathematics, certain infinite series can indeed have a finite sum, provided they meet specific convergence criteria.

One important type of infinite series is the infinite geometric series. As noted, such a series will converge to a finite sum if the absolute value of the common ratio is less than 1. This makes it possible to calculate the sum of an infinite number of terms using a simple formula. The notion of an infinite series is not merely a theoretical concept; it has practical applications in various areas, including engineering, science, and finance.

The sum of the specific infinite geometric series given in the exercise, \( \frac{-10}{1 - 0.2} \), is a solid example, converging to -12.5 because the series meets the criteria for convergence, with a common ratio less than 1. Without understanding the principle behind infinite series, one might never realize that adding an infinite number of these terms results in such a straightforward and finite answer.