91Ó°ÊÓ

In the context of a geometric series, convergence refers to the behavior of the series as the number of terms grows indefinitely. For a geometric series to converge, the absolute value of the common ratio \(r\) must be less than 1, written as \(|r|<1\). This condition ensures that as we add more and more terms to the series, the series approaches a definite value rather than fluctuating towards infinity or diverging.

Understanding why convergence happens can be intuitive if you think of each term in a series becoming increasingly smaller. As terms diminish toward zero, the series stabilizes around a specific sum. In the provided exercise, the common ratio \(r\) was found to be \(\frac{1}{2}\), which is clearly less than 1, indicating that the series converges.

The common ratio is a crucial element in geometric series. It's the factor by which we multiply any term of the series to get the next term. Determining the common ratio is the first step in analyzing a geometric series. In the given exercise, the series starts with terms like \(\frac{5}{2^2}, \frac{5}{2^3}, \frac{5}{2^4}, \ldots\) and the common ratio \(r\) is calculated as:

• The ratio of the second term to the first: \(\frac{\frac{5}{2^3}}{\frac{5}{2^2}} = \frac{2^2}{2^3} = \frac{1}{2}\)

This constant multiplication factor \(\frac{1}{2}\) is what transforms each term into the next, creating a predictable series pattern.

Recognizing the common ratio helps establish whether a series converges or diverges, and is essential for calculating the sum of an infinite series.

Once we know whether a geometric series converges, we can calculate its sum if it does. The sum of an infinite geometric series is given by the formula:

• \(S = \frac{a}{1 - r}\)

Here, \(S\) represents the sum of the series, \(a\) is the first term, and \(r\) is the common ratio. This formula simplifies the complex task of summing infinite terms to a simple arithmetic expression.

In the original exercise, the first term \(a\) was \(\frac{5}{4}\) (or when \(m = 2\)), and the common ratio \(r\) was determined to be \(\frac{1}{2}\). Plugging these into the formula, the series sum is:

• \(S = \frac{\frac{5}{4}}{1 - \frac{1}{2}} = \frac{\frac{5}{4}}{\frac{1}{2}} = \frac{5}{2}\)

Understanding the sum of series concept allows one to capture the overall value that an infinite series approaches, providing insight into the series' behavior.