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A geometric series is a special type of infinite series where each term after the first is found by multiplying the previous one by a constant. This constant is known as the common ratio, denoted as \( r \). Geometric series can be written in the form:

• \( a, ar, ar^2, ar^3, ... \)

where \( a \) is the first term, and \( r \) is the common ratio. For example, in the series \( 2, 4, 8, 16, ... \), the first term \( a \) is 2, and the common ratio \( r \) is 2 because each term is twice the previous term.

Geometric series have interesting properties. If \( |r| < 1 \), the series \( \sum_{n=0}^{\infty} ar^n \) converges to a sum \( \frac{a}{1-r} \). If \( |r| \geq 1 \), the series diverges. Hence, the behavior of the series is determined mostly by the absolute value of the common ratio \( r \). This makes determining convergence straightforward in such series.

An infinite series is a sum involving an infinite number of terms. It is written as \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) represents the general term of the series. Unlike a finite series that has a limited number of terms, an infinite series extends infinitely. The challenge with infinite series is deciding whether their sum reaches a finite limit or not as more terms are added.

In the context of math, two primary behaviors are associated with infinite series:

• Convergence: An infinite series converges if the sum of its terms approaches a specific finite value as more and more terms are added.
• Divergence: If the sum does not approach a specific finite limit, the series is said to diverge.

The convergence or divergence of an infinite series can often be determined using specific tests such as the geometric series test, ratio test, or other convergence tests, depending on the form and structure of the series involved.

Convergence criteria are rules and tests used to determine if an infinite series converges. For a series to converge, its terms must decrease in such a manner that their sum reaches a specific value as the number of terms increases.

Geometric series have straightforward convergence criteria. Specifically, a geometric series \( \sum_{n=0}^{\infty} ar^n \) converges under the condition that \( |r| < 1 \). This is because, intuitively, the terms of the series get smaller and smaller, ensuring that their total forms a finite value. On the other hand, if \( |r| \geq 1 \), the terms either stay large or oscillate, meaning their total does not settle on a particular value, leading to divergence.

In the exercise example, the series \( \sum_{n=1}^{\infty} \left( \frac{-2}{3} \right)^n \) is assessed using the geometric series test. We detect the common ratio \( r = \frac{-2}{3} \) and determine its absolute value \( |r| = \frac{2}{3} \). Since \( \frac{2}{3} < 1 \), the series is convergent based on this convergence criterion. Understanding and applying the right test provides a structured way to analyze the behavior of infinite series.