91Ó°ÊÓ

In the world of series, particularly geometric series, convergence is a critical concept to understand. A series is said to converge if it approaches a specific value as we sum progressively more terms. For a geometric series, which takes the form \( \sum_{k=0}^{\infty} ar^k \), convergence depends on the common ratio, \( r \).
The key condition for the convergence of an infinite geometric series is that the absolute value of the common ratio must be less than one:

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

In our specific problem, the common ratio \( r \) is \( \frac{2}{3} \). Since \( \left|\frac{2}{3}\right| = \frac{2}{3} < 1 \), the series converges. Being aware of this condition helps us easily find whether an infinite geometric series will settle on a particular sum or not.

Once we've determined that a geometric series converges, we can calculate its sum. The sum of a convergent infinite geometric series \( \sum_{k=0}^{\infty} ar^k \) is calculated using the formula: \[ \text{Sum} = \frac{a}{1-r} \] where \( a \) is the first term of the series, and \( r \) is the common ratio.
In the exercise, we identified \( a = \frac{4}{9} \) and \( r = \frac{2}{3} \), leading us to the sum: \[ \frac{\frac{4}{9}}{1-\frac{2}{3}} = \frac{\frac{4}{9}}{\frac{1}{3}} = \frac{4}{3} \] The beauty of this is that we can find the sum of infinitely many terms in a series by using this simple formula, provided the series converges.

The concept of a common ratio is essential in understanding geometric series. The common ratio, denoted \( r \), is the factor by which we multiply each term to get the next one in the series. This ratio remains constant throughout the series.
To identify \( r \), we look at the expression for the series: each term is \( ar^k \). For our given series, we rewrote the general term \( \left(\frac{2}{3}\right)^{k+2} \) to reveal the form \( \left(\frac{4}{9}\right)\left(\frac{2}{3}\right)^k \), showing that \( r = \frac{2}{3} \).

• The common ratio determines whether the series converges.
• The magnitude of \( r \) affects the speed at which the series sum settles into its final value if it converges.

Understanding the common ratio helps demystify why certain series sum up or continue growing indefinitely.