91Ó°ÊÓ

In a geometric series, each subsequent term is derived from the previous one by multiplying by a fixed number, known as the common ratio. For the series we are discussing, the common ratio can be found by dividing one term by the previous term. Specifically, if you look at the second term of our series, \( \frac{25}{6} \), and divide it by the first term, \( -\frac{125}{36} \), you get \( -\frac{36}{5} \). It’s essential to check this ratio with more than one pair of terms to confirm it remains consistent. In our example, dividing the third term, \(-5\), by the second term also yields \(-\frac{36}{5}\), confirming it as the common ratio.
Understanding the common ratio is crucial as it determines the behavior and properties of the series, namely whether the series can converge to a finite number or not.

A convergent series is one where the sums of its terms approach a specific value as you keep adding more terms. For geometric series, they converge only if the absolute value of the common ratio is less than 1; mathematically, this is expressed as \(|r| < 1\).
Think of it like a balancing act – if the common ratio's absolute value is smaller than 1, the terms get progressively smaller, allowing the series to settle down to a finite sum. However, if the ratio's absolute value equals or exceeds 1, the series terms do not shrink to zero, and the series is termed divergent.
In the series \(-\frac{125}{36}+\frac{25}{6}-5+6-\cdots \), the common ratio is \(-\frac{36}{5}\), whose absolute value is greater than 1. As a result, this series is divergent, meaning it does not converge to any finite value.

The geometric series sum provides the total of all terms in a geometric progression. For an infinite geometric series that is convergent, the formula to find the sum \(S\) is \( S = \frac{a}{1-r} \), where \(a\) is the first term and \(r\) is the common ratio.
In cases where \(|r| < 1\), this formula can be used to determine the series' sum because the terms eventually become negligible, and you end up with a finite number.
However, if the series is divergent, like in our example with \(-\frac{125}{36}+\frac{25}{6}-5+6-\cdots\), where \(|r|\) is not less than 1, the sum isn't defined. Divergent series don't sum to a finite value, so the sum formula doesn’t apply. It’s important to establish convergence first before using this formula to calculate the series sum.