91Ó°ÊÓ

A geometric series is a fascinating mathematical concept where each term is obtained by multiplying the previous term by a constant, known as the common ratio. In our given exercise, the series is represented by \( \sum_{n=1}^{\infty} 6(0.9)^{n-1} \). This type of series can be easily identified by noting the repeating pattern in its terms.

• The first term \( a \) is \( 6 \).
• The common ratio \( r \) is \( 0.9 \).

A geometric series can either converge or diverge depending on the value of \( r \). If \( |r| < 1 \), the series will converge, meaning it will approach a specific value as the number of terms increases indefinitely. In contrast, if \( |r| \geq 1 \), the series diverges, implying it does not settle towards any particular value. Understanding these properties allows us to predict the behavior of different geometric series.

The convergence of a series is a key concept in understanding its nature. For geometric series, there's a straightforward test to determine whether the series converges. Consider the exercise again, where the common ratio \( r \) is \( 0.9 \). To apply the test:

• Check if the absolute value of the common ratio \( |r| \) is less than 1.

For the series \( \sum_{n=1}^{\infty} 6(0.9)^{n-1} \), we find \( |0.9| = 0.9 \), which is indeed less than 1. This implies the series is convergent.

By applying this simple test, you can quickly determine whether a geometric series will sum up to a finite value or not. This is crucial when working with infinite series, especially in fields such as calculus and analysis, as it simplifies the understanding of more complex mathematical concepts.

Once we've determined that a geometric series is convergent, the next step is to find its sum. This is possible through a neat formula. For a geometric series \( \sum_{n=0}^{\infty} ar^n \), if it converges, its sum \( S \) is given by:

\[S = \frac{a}{1 - r}\]
Applying this formula to our exercise's series, where \( a = 6 \) and \( r = 0.9 \), we substitute these values:

• \( S = \frac{6}{1 - 0.9} \)
• \( S = \frac{6}{0.1} \)
• \( S = 60 \)

Thus, the series converges to \( 60 \). This formula provides a quick way to find the sum of any convergent geometric series, making it an invaluable tool for mathematicians and learners alike.