Sum of an Infinite Geometric Sequence

Determine the sum of the infinite geometric sequence represented by $8 \left( \frac{2}{3} \right)^{n-1}$.

The problem of finding the sum of an infinite geometric sequence is a classic problem in discrete mathematics and serves as an essential introduction to series and sequences. A geometric sequence is one where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. When the absolute value of this common ratio is less than one, the sequence has a finite sum, which is given by the formula $S = \frac{a}{1 - r}$, where 'a' is the first term of the sequence and 'r' is the common ratio.

In this problem, understanding the conditions under which the series converges is crucial. We must confirm that the absolute value of the common ratio is indeed less than one to ensure the series sums to a finite number. Convergence of infinite series is a foundational concept that finds applications across various domains of mathematics and computer science, such as algorithms and computational complexity.

Beyond the direct application to sequences, this problem also provides a framework for understanding more complex topics such as generating functions and their relationship to sequences, which are key tools in solving recurrence relations and analyzing series that are common in computer science, especially in algorithm design and data analysis.