91Ó°ÊÓ

A series is a sum of terms that follow a specific pattern. In mathematics, a series can either converge or diverge. When we say a series is convergent, it means that the sum of all its terms approaches a specific finite value as more terms are added.
A geometric series is an example where each term is a constant multiple of the previous term. In the case of convergence, for a geometric series to be convergent, the absolute value of the common ratio must be less than 1.
For the given series: \[1 + \frac{1}{6} + \frac{1}{6^2} + \frac{1}{6^3} + \frac{1}{6^4} + \tiny\text{other terms}ormalsize\] the common ratio \(\frac{1}{6}\) is indeed less than 1. Therefore, we can conclude that our series is convergent.

In a geometric series, the pattern of the terms is determined by the common ratio. The common ratio is calculated by dividing any term in the series by the previous term. For our geometric series: \[1 + \frac{1}{6} + \frac{1}{6^2} + \frac{1}{6^3} + \frac{1}{6^4} + \tiny\text{other terms}ormalsize\] we can see that the term \(\frac{1}{6}\) comes right after 1. Dividing \(\frac{1}{6} \text{by} 1 = \frac{1}{6}\), we find the common ratio is \(\frac{1}{6}\). Understanding the common ratio is key while working with geometric series because it determines both the behavior and the sum of the series.

For finding the sum of a convergent geometric series, we use a special sum formula. This formula allows us to calculate the total sum of an infinite number of terms in the series. The formula is: \[S = \frac{a}{1 - r}\] where 'S' is the sum of the series, 'a' is the first term, and 'r' is the common ratio.
In our case, the first term 'a' is 1, and the common ratio 'r' is \(\frac{1}{6}\). Substituting these values into the formula, we get:\[\frac{1}{1 - \frac{1}{6}} = \frac{1}{\frac{5}{6}} = \frac{6}{5}\] Thus, the sum of the series is \(\frac{6}{5}\). By using the sum formula, you can easily find the sum of any convergent geometric series once you know the first term and the common ratio.