91Ó°ÊÓ

A geometric series is a sequence of terms where each term is found by multiplying the previous term by a constant called the common ratio. This constant ratio, denoted as \( r \), should always be the same between consecutive terms.
For example, the series \( \frac{1}{5}+\frac{1}{5^{4}}+\frac{1}{5^{7}}+\frac{1}{5^{10}}+\frac{1}{5^{13}}+\text{...} \) has a common ratio of \( \frac{1}{125} \). Each term can be obtained by multiplying the previous term by this ratio.
The general form of a geometric series is given by:
\[ a + ar + ar^2 + ar^3 + \text{...} \] where:

• \( a \) is the first term
• \( r \) is the common ratio
• Each subsequent term is obtained by multiplying the preceding term by \( r \)

This series continues indefinitely if it is infinite. Understanding the common ratio is crucial to solving problems related to geometric series.

Convergence in the context of a series refers to whether the series approaches a finite value as the number of terms goes to infinity. For geometric series, this depends primarily on the common ratio \( r \).
A geometric series converges if the absolute value of the common ratio \( r \) is less than 1. Mathematically, this is written as:
\[ |r| < 1 \] If \( |r| \) is less than 1, the terms become smaller and smaller, and their sum approaches a finite value. If \( |r| \) is equal to or greater than 1, the series does not converge, meaning the sum of its terms does not approach a specific value and can become infinitely large.
In our example series \( \frac{1}{5}+\frac{1}{5^{4}}+\frac{1}{5^{7}}+\frac{1}{5^{10}}+\frac{1}{5^{13}}+\text{...} \), the common ratio is \( \frac{1}{125} \). Since this is less than 1, we can say the series is convergent. This means it will sum up to a finite number, which can be calculated.

Once we establish that a geometric series is convergent, we can find its sum using a specific formula. For an infinite geometric series where \( |r| < 1 \), the sum \( S \) is given by:
\[ S = \frac{a}{1-r} \] Here:

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

To find the sum of the series \( \frac{1}{5}+\frac{1}{5^{4}}+\frac{1}{5^{7}}+\frac{1}{5^{10}}+\frac{1}{5^{13}}+\text{...} \):

• First term \( a = \frac{1}{5} \)
• Common ratio \( r = \frac{1}{125} \)
• Formula: \( S = \frac{1/5}{1 - 1/125} \)

Plugging in the values, we simplify:
\[ S = \frac{\frac{1}{5}}{1 - \frac{1}{125}} = \frac{\frac{1}{5}}{\frac{124}{125}} = \frac{1}{5} \times \frac{125}{124} = \frac{125}{620} = \frac{25}{124} \]
Therefore, the sum of the given geometric series is \( \frac{25}{124} \).