91Ó°ÊÓ

When dealing with infinite geometric series, a key factor in determining whether the series will have a sum is the concept of convergence. A series is said to converge when it approaches a certain value as the number of terms increases indefinitely. To check for convergence, we use the common ratio, represented by \( r \).
For an infinite geometric series to converge, the absolute value of the common ratio must be less than 1, that is, \( |r| < 1 \). When this condition is met, the series has a finite sum. In our given series \(1.5 + 0.015 + 0.00015 + \cdots\), we have a common ratio \(r = 0.01\), which satisfies \(|0.01| < 1\).
Therefore, this series converges, meaning there is a fixed sum to infinity which can be calculated.

The geometric series formula is crucial in finding the sum of an infinite series that converges. For any infinite geometric series where the absolute value of the common ratio \( |r| < 1 \), the sum \( S \) can be calculated using the formula:

• \( S = \frac{a}{1-r} \)

Here, \( a \) is the first term of the series.
In the example provided, the first term \( a \) is 1.5, and the common ratio \( r \) is 0.01.
By substituting these values into the formula, we have:

• \( S = \frac{1.5}{1 - 0.01} = \frac{1.5}{0.99} \)

This calculation provides us with the sum of the series, showing how even an infinite series can produce a finite result when it converges.

The common ratio is a fundamental component in any geometric series. It defines how each term in the series relates to the previous term. You find the common ratio \( r \) by dividing any term by its preceding term in the series.
For the series at hand, the first term is 1.5, and the second term is 0.015. We calculate the common ratio as follows:

• \( r = \frac{0.015}{1.5} = 0.01 \)

The value of the common ratio is significant because it not only informs us whether a series converges but also influences how the terms of the series decrease or increase. When \( r \) is a fraction (i.e., \(|r| < 1\)), each successive term gets smaller, leading the series to potentially converge to a sum, as demonstrated above.