91Ó°ÊÓ

When we talk about the convergence of a series, we're investigating whether the series approaches a certain value as we keep adding more terms. This is especially important for infinite series, where adding infinite terms could lead to a sum that is undefined or infinite. In the context of geometric series, a simple rule helps us determine if the series converges: check the absolute value of the common ratio, denoted as \( |r| \).
If \( |r| < 1 \), the series converges; otherwise, it diverges. This happens because a common ratio smaller than 1 in absolute value ensures that each successive term gets closer to zero, making the whole series sum to a finite value. Let's consider our example:

• The series is: \( 27 - \frac{27}{2} + \frac{27}{4} - \frac{27}{8} + \frac{27}{16} - \cdots \)
• The common ratio \( r = -\frac{1}{2} \) has an absolute value of \( \frac{1}{2} \), which is less than 1.
• Thus, the series converges.

To summarize, understanding convergence is pivotal in determining if an infinite geometric series has a meaningful sum.

Once we confirm that a geometric series is convergent, the exciting part emerges: calculating its sum. The formula for the sum \( S \) of a convergent infinite geometric series is:

• \( S = \frac{a}{1 - r} \)

where \( a \) is the first term and \( r \) is the common ratio. This formula stems from summing a geometrically shrinking sequence, which, when summed to infinity, approaches a finite limit.
In our specific series:

• The first term \( a = 27 \)
• The common ratio \( r = -\frac{1}{2} \)

Plugging these values into the formula gives:

• \( S = \frac{27}{1 - (-\frac{1}{2})} = \frac{27}{1 + \frac{1}{2}} = \frac{27}{\frac{3}{2}} = 18 \)

Thus, if the series converges, we can confidently calculate its sum using this formula, giving us a succinct measure of the entire infinite series.

The common ratio in a geometric series is a fundamental concept that defines how the series progresses. It's the constant factor by which we multiply each term to get the next term.
Mathematically, to find the common ratio \( r \), divide any term in the geometric series by the preceding term. Consider our example series:

• First term \( a = 27 \)
• Second term is \( -\frac{27}{2} \)
• To find \( r \), we compute \( r = \frac{-\frac{27}{2}}{27} = -\frac{1}{2} \)

This common ratio of \(-\frac{1}{2}\) tells us that each term is half the size of the preceding one, and moreover, alternates in sign.
Understanding the common ratio is crucial because:

• It helps determine if the series converges or diverges.
• Aids in calculating the sum when the series is convergent.

Thus, the common ratio is not just a number—it encapsulates the entire behavior and direction of the series.