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A geometric series is a special type of series where each term is obtained by multiplying the previous term by a fixed number, called the common ratio. The general form of a geometric series is given by:

• The first term is represented as \( a \).
• The common ratio, denoted as \( r \), is the fixed number multiplied to get subsequent terms.

For example, the series with terms \( a, ar, ar^2, ar^3, \ldots \) is a geometric series with a common ratio \( r \).
A geometric series converges if the absolute value of the common ratio is less than 1, that is, \( |r| < 1 \). Its sum can be calculated using the formula:\[S = \frac{a}{1 - r}\]Because geometric series have a predictable pattern, they are easier to solve than some other types of series.

When studying series, determining whether a series converges or diverges is crucial. Convergence tests provide tools to analyze the behavior of series.
For geometric series, the convergence test focuses on the common ratio \( r \):

• If \( |r| < 1 \), the series converges and has a definite sum.
• If \( |r| \geq 1 \), the series diverges and does not approach a finite value.

Besides geometric series specifics, other tests like the Comparison Test, Ratio Test, and Root Test provide methods to determine convergence for more complex series. However, for many problems, especially involving geometric series, checking the common ratio is a clear and efficient strategy.

An infinite series is a sum of infinitely many terms, written in the form \( \sum_{n=0}^{ \infty} a_n \), where \( a_n \) represents each term. A crucial aspect of an infinite series is whether it converges to a finite sum as more terms are added.
In the realm of mathematics, we often ask if the infinite sum settles to a particular value, known as convergence, or if it continues to grow indefinitely, known as divergence.
Geometric series are a subset where the terms form a geometric progression. If the conditions for convergence are met, they provide an elegant example of how an infinite number of terms can add up to a finite sum. This characteristic makes them an important tool in mathematical analysis.

Finding the sum of a series, especially when it's infinite, requires particular techniques and strategies. For a convergent geometric series, the sum can be calculated using:\[S = \frac{a}{1 - r}\]where \( a \) is the first term and \( r \) is the common ratio with \( |r| < 1 \).
For the series \( \sum_{n=0}^{ \infty} \frac{2^n - 1}{3^n} \), the sum is determined by breaking it into two separate geometric series:

• \( \sum_{n=0}^{ \infty} \left( \frac{2}{3} \right)^n \), yielding 3.
• \( \sum_{n=0}^{ \infty} \left( \frac{1}{3} \right)^n \), yielding \( \frac{3}{2} \).

Thus, the sum of the original series is \( 3 - \frac{3}{2} \), resulting in \( \frac{3}{2} \). Understanding how to find the sum is key to solving problems involving infinite series.