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A geometric series is a type of series where each term is derived from the previous one by multiplying it by a constant factor, called the "common ratio." This kind of series follows the form \(\sum a \cdot r^{n}\), where \(a\) represents the first term of the series, \(r\) is the common ratio, and \(n\) starts from 0 and goes to infinity.

A classical example of a geometric series is \(2 + 2r + 2r^2 + 2r^3 + \ldots\).

Things to remember about geometric series:

• The first term, denoted as \(a\), remains constant throughout the series.
• The common ratio \(r\) determines the behavior and pattern of the series.
• Understanding these two components helps greatly in determining whether the series converges or diverges.

The convergence of a geometric series is dependent on its common ratio. The rule is quite simple: a geometric series converges if the absolute value of the common ratio \(r\) is less than 1, that is, \(-1 < r < 1\).

If the series converges, it approaches a finite limit as more terms are added. The sum of such an infinite geometric series can be calculated as \(S = \frac{a}{1-r}\), where \(S\) is the sum, \(a\) is the first term, and \(r\) is the common ratio.

• A value of \(|r| < 1\) leads to convergence and an overall finite sum.
• It’s a quick and efficient way to determine the behavior of the series.

However, if \(r\) falls outside of this range, the series does not settle around a specific number and is known as divergent.

When a series is classified as divergent, it means that as you add an infinite number of terms, there is no fixed sum. In other words, the series doesn't stabilize to a finite value as more and more terms are added. A geometric series with a common ratio \(r\) that is equal to or greater than 1, or less than -1, will always be divergent.

For this specific exercise, the series \(\sum_{n=0}^{\infty} 2(-1.03)^{n}\) is divergent since the common ratio \(r = -1.03\) falls outside the convergence range \(-1 < r < 1\).

Key points about divergent series include:

• The series does not sum to a single, finite limit.
• The value of \(r\) determines the divergence - in this case, being greater than 1 in absolute value leads to divergence.

Hence, understanding the divergence of a series is crucial in analysis and helps predict the behavior of series when summing infinitely many terms.