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Abstraction is a core concept in mathematics that allows for simplification of complex ideas by removing excess information and focusing on what is necessary for a particular problem or pattern. In the case of infinite geometric series, abstraction enables mathematicians to represent an infinitely long list of numbers with a concise expression. By abstracting the details of each individual term, we can view the series as a whole and notice patterns that lead to general formulas.

For instance, the series in the exercise, which seems to go on without end, can be symbolized abstractly as a sequence of terms involving the variable 'r'. Each consecutive term is produced by multiplying the previous one by 'r'. This abstract view allows us to use a formula to represent something seemingly complex like an infinite series in a surprisingly simple way - a single mathematical expression.

The sum of a geometric series is derived from its pattern of multiplication by a common ratio. Specifically, the sum of an infinite geometric series can be calculated by the formula \(S = \frac{a}{1-r}\) where 'a' is the first term and 'r' is the common ratio, under the condition that \(|r| < 1\).

The idea behind this formula is rooted in the concept of convergence, where the infinite addition of terms eventually approaches a finite value. In the given exercise, the sum of the series is represented as \(\frac{1}{1-r}\), which tells us that even though we're adding an infinite number of terms, as long as the common ratio is between -1 and 1, we will end up with a finite sum.

Mathematical problem-solving encompasses understanding the problem, devising a plan, carrying out the plan, and then looking back to reflect on the solution. This process was evident in the step-by-step approach to finding the sum of the infinite geometric series.

Understanding involved recognizing the series and its components. Devising a plan meant applying the known geometric series sum formula, which is the abstraction of an infinite process into a manageable form. Execution was straightforward: substituting values into the formula. Finally, looking back included verifying the conditions for convergence and ensuring the common ratio allowed for a sum to exist. By teaching problem-solving strategies alongside exercises, students develop the critical skill to tackle diverse mathematical challenges.