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In an infinite geometric series, the common ratio is a key parameter that dictates the behavior of the series. The common ratio is found by dividing any term in the series by its preceding term. For example, given the series 1.2, -2.4, 4.8, -9.6, ..., the common ratio is calculated as \[ \frac{-2.4}{1.2} = -2. \] This value signifies how each term relates to the previous one.

If every term is multiplied by a common ratio of -2 to get to the next term, this regular interval of multiplication helps us identify the nature of the series. Understanding and determining the common ratio is crucial for further analysis, such as checking convergence and calculating the series sum.

The concept of convergence is essential when working with infinite geometric series. For an infinite geometric series to converge (meaning it approaches a finite sum as the number of terms goes to infinity), the absolute value of the common ratio must be less than 1. This can be formulated as \[ \left| r \right| < 1. \]

When a series converges, the terms decrease in magnitude and approach zero, leading the sum to stabilize at a particular value. In the provided exercise, the common ratio was found to be -2, which means \[ \left| -2 \right| = 2, \] a value greater than 1. Therefore, this series does not converge as the terms grow larger in magnitude, causing the series to diverge rather than sum to a finite value.

Calculating the sum of an infinite geometric series can only be done if the series converges. When convergence criteria are met \[ \left| r \right| < 1, \] the sum can be calculated using the formula: \[ S = \frac{a}{1-r}, \] where \[ S \] is the sum of the series, \[ a \]is the first term, and \[ r \] is the common ratio.

However, as seen in the given exercise with the series 1.2, -2.4, 4.8, -9.6, ..., since the common ratio \[ -2 \] is not within the desired range (\[ \left| -2 \right| = 2 \] > 1), the series does not converge. Consequently, this series does not have a sum. Understanding the conditions under which an infinite geometric series sums up helps in correctly evaluating whether specific series can be summed.