91Ó°ÊÓ

In mathematics, understanding whether a series converges or diverges is crucial for determining if the sum of the series can be calculated. For geometric series, convergence depends largely on the common ratio, denoted as \( r \). Geometric series have terms that follow the form \( ar^k \), where \( k \) is a non-negative integer starting from 0.
To establish convergence, we check the absolute value of the common ratio:

• If \( |r| < 1 \), the series converges. This means the infinite series approaches a fixed sum as the number of terms tends to infinity.
• If \( |r| \geq 1 \), the series diverges, and no finite sum exists.

In our example, the common ratio \( r = \frac{2}{7} \) displays that \( |r| < 1 \), confirming convergence. Therefore, this geometric series will reach a specific sum despite having an infinite number of terms.

The common ratio \( r \) in a geometric series is the factor by which each term progresses from the previous one. To find the common ratio, take the ratio of any term to its predecessor.
Mathematically, for a sequence \( a, ar, ar^2, ar^3, \ldots \), \( r \) is constant such that:

• \( r = \frac{successive\ term}{previous\ term} \)

In the given problem, the sequence is \( 1, \frac{2}{7}, \left( \frac{2}{7} \right)^2, \left( \frac{2}{7} \right)^3, \ldots \)
Here, \( r = \frac{2}{7} \), a number significantly less than 1, ensuring each term is smaller than the last one. Recognizing and correctly determining the common ratio is essential for evaluating both convergence and the sum of the series.

Once a geometric series is confirmed to converge, the sum of all its infinite terms can be calculated using a formula. This is especially practical in series with a diminishing pattern where each subsequent term decreases significantly due to the nature of \( |r| < 1 \).
The formula for the sum \( S \) of an infinite geometric series is:

• \( S = \frac{a}{1 - r} \)

where \( a \) is the first term and \( r \) is the common ratio.
In our exercise, with \( a = 1 \) and \( r = \frac{2}{7} \), the sum is computed as:

• \( S = \frac{1}{1 - \frac{2}{7}} = \frac{1}{\frac{5}{7}} = \frac{7}{5} \)

Thus, the result \( \frac{7}{5} \) signifies the total sum to which the series converges. The formula helps quantify the infinite series into a single finite sum, illustrating the concept of convergence in practical terms.