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A geometric series is a sequence of terms where each term after the first is found by multiplying the previous term by a constant called the common ratio. When we talk about the 'sum of a geometric series,' we're interested in finding the total value when adding all terms together.
An important concept is whether this series has a finite sum, especially if it goes on infinitely. This depends on the properties of the series, particularly the common ratio.
In our example, the series is: \[1 + \frac{1}{2^{3}} + \frac{1}{2^{6}} + \frac{1}{2^{9}} + \frac{1}{2^{12}} + \cdots \]
We identified the sum of this geometric series using a special formula for infinite series that converge. Convergence here means the sum approaches a certain value as more terms are added.

The 'common ratio' is a key component of a geometric series. To find it, you divide one term by the term that directly precedes it.
In our series: \(1 + \frac{1}{2^3} + \frac{1}{2^6} + \frac{1}{2^9} + \frac{1}{2^{12}} + \cdots\), the first term (\(a\)) is 1. To find the 'common ratio' \(r\), we divide the second term by the first term: \[r = \frac{\frac{1}{2^3}}{1} = \frac{1}{8}\]
This shows us that each term is obtained by multiplying the previous term by \(\frac{1}{8}\). This ratio stays the same throughout the entire series, which is what makes it a 'geometric' series.

The formula for the sum of an infinite geometric series that converges is crucial for solving our problem. The formula is: \[S = \frac{a}{1 - r}\] where:

• \(S\) is the sum of the series
• \(a\) is the first term of the series
• \(r\) is the common ratio

In our problem, we determined that \(a = 1\) and \(r = \frac{1}{8}\). Plugging these values into the formula gives us: \[S = \frac{1}{1 - \frac{1}{8}} = \frac{1}{\frac{7}{8}} = \frac{8}{7}\]
This shows that the sum of the geometric series is \(\frac{8}{7}\).

Convergence is an important concept when dealing with infinite series. A series converges if its terms approach a specific value as more terms are added.
For a geometric series to converge, the absolute value of the common ratio \(r\) must be less than 1 (\(\left| r \right| <1\)).
In our example, we found that \(r = \frac{1}{8}\). Since \(\left| \frac{1}{8} \right| < 1\), the series converges.
This means that as we keep adding more terms of the series \(1 + \frac{1}{2^{3}} + \frac{1}{2^{6}} + \frac{1}{2^{9}} + \frac{1}{2^{12}} + \cdots\), the sum will approach a definite value, which we calculated to be \(\frac{8}{7}\).