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An infinite series is essentially a sum of an infinite number of terms. In our specific case, we are looking at the series \( \sum_{n=0}^{\infty} (-2/3)^n \). This is an example of what mathematicians call a geometric series. Each term in the series becomes smaller and smaller. But since there are infinitely many terms, you might wonder whether the series adds up to a finite number or shoots off to infinity.

Understanding infinite series can be quite fascinating. The main question mathematicians often explore is whether such series converges, meaning it approaches a specific number as you keep adding terms. If it converges, the series isn't just a diverging mess—it has a meaningful value in mathematical analysis. This concept is vital in calculus, analysis, and even in real-world applications like physics and engineering.

To know if our infinite series \( \sum_{n=0}^{\infty} (-2/3)^n \) sums up to a finite number, we use something called the convergence criteria. For geometric series specifically, a simple rule applies: the series converges if the absolute value of the common ratio, \( |r| \), is less than 1.

In our problem, the common ratio \( r \) is \(-2/3\). The absolute value \( |-2/3| \) is \( 2/3 \), which is indeed less than 1. Hence, we confirm that the series converges. This means the infinite series actually has a finite sum, which is pretty neat!

• If \( |r| < 1 \), the series converges.
• If \( |r| \geq 1 \), the series diverges.

Once we know the series converges, it's time to calculate the actual sum using the sum formula for an infinite geometric series. This formula is a handy tool to quickly find the finite sum of a converging series.

The sum \( S \) of an infinite geometric series is given by:\[ S = \frac{a}{1 - r}\]where \( a \) is the first term and \( r \) is the common ratio. For our series, \( a = 1 \) since \( (-2/3)^0 = 1 \), and \( r = -2/3 \). Substituting these values into the formula, we get:\[ S = \frac{1}{1 - (-2/3)} = \frac{1}{1 + 2/3} = \frac{1}{5/3}\]

The common ratio is a key component in understanding and solving geometric series. For the series \( \sum_{n=0}^{\infty} (-2/3)^n \), the common ratio \( r \) is \(-2/3\). This ratio tells you how each term relates to the previous one.

Each term is obtained by multiplying the previous term by \(-2/3\). So, if you start with the first term \( a = 1 \), the next few terms would be \((r \cdot 1), (r^2 \cdot 1), (r^3 \cdot 1)\), and so on.

• The common ratio plays a crucial role in determining the behavior of the series, specifically its convergence or divergence.
• A ratio with absolute value less than 1 indicates the series converges while greater than or equal to 1 suggests the series diverges.

Understanding the common ratio provides a clear view of how the geometric series' terms shrink as they progress, making it easier to anticipate whether the series converges or not.