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Understanding the convergence of an infinite geometric series is crucial for determining its sum. Precisely, a series converges when the sum approaches a finite number as the number of terms goes to infinity. For a geometric series with a common ratio 'r', this happens if the absolute value of 'r' is less than 1, that is, \(|r| < 1\).

This condition arises from the fact that as we keep multiplying a number less than 1 by itself, the result gets smaller and closer to zero. Conversely, if the absolute value of the common ratio is equal to or greater than 1, the series does not converge because the terms do not diminish as they progress; rather, they either stay constant (when \(|r| = 1\)) or increase without bound (when \(|r| > 1\)). This ensures that their sum grows indefinitely, known as divergence.

The common ratio in a geometric series is the factor by which each term is multiplied to obtain the next term. It is denoted as 'r' and can be any real number. This ratio is a cornerstone in the behavior of geometric series, determining whether the series converges or diverges.

To find the common ratio, divide any term in the series by the previous term (except for the first term, which has no preceding term). For example, in the series 2, 4, 8, 16, ..., the common ratio is 2 because each term is twice its predecessor. The magnitude of this ratio helps us understand how quickly the terms of the series are growing or shrinking.

Once you establish that an infinite geometric series converges, calculating its sum becomes feasible using the geometric series formula. The formula is \(S = \frac{a}{1 - r}\), where 'S' represents the sum of the series, 'a' is the first term, and 'r' is the common ratio.

As an illustration, consider a series with first term 5 and common ratio 0.5. The sum is obtained by substituting these values into the formula resulting in \(S = \frac{5}{1 - 0.5} = 10\). This formula is derived from the sum of a finite geometric series and the concept that as the number of terms grows infinitely, the portion of the formula accounting for the remaining terms approaches zero, leaving us with this simple expression. It's a powerful tool, compactly summarizing a potentially infinite process.

While some geometric series have the pleasant property of convergence, others may diverge. This occurs when the terms do not approach zero, which means the sum of the terms cannot settle at a finite value. Specifically, a geometric series will diverge if its common ratio 'r' fulfills one of these conditions: \(|r| = 1\) or \(|r| > 1\).

For instance, the series 3 + 3 + 3 + ... has a common ratio of 1. Each term is the same, and naturally, adding up identical non-zero numbers an infinite number of times leads to infinity. Similarly, a series with a common ratio greater than 1, say 2, like 1, 2, 4, 8, ..., doubles each term. Therefore, the series expands unboundedly, illustrating the concept of divergence within geometric series.