Explain how to tell if a geometric series is convergent or divergent. Include an example of a convergent series and an example of a divergent series in your explanation.
To determine if a geometric series is convergent or divergent, you need to examine the common ratio (r) of the series. A geometric series is defined as the sum of the terms of a geometric sequence.
The general form of a geometric sequence is: a, ar, ar^2, ar^3, ...
The formula for the sum of a geometric series (S) is given by:
S = a / (1 - r)
Now, let's discuss how we can determine if a geometric series is convergent or divergent:
1. Convergent series: A geometric series is convergent if the absolute value of the common ratio (|r|) is less than 1. In other words, if |r| < 1, the series converges to a finite value.
Example: Let's consider the geometric series with the first term (a) of 2 and a common ratio (r) of 1/2.
The series will be: 2, 1, 1/2, 1/4, 1/8, ...
The absolute value of the common ratio is |1/2| = 1/2, which is less than 1. Therefore, this series is convergent.
To find the sum, we can use the formula:
S = a / (1 - r)
Substituting the values: S = 2 / (1 - 1/2) = 2 / (1/2) = 4
So, the sum of this series is 4.
2. Divergent series: A geometric series is divergent if the absolute value of the common ratio (|r|) is greater than or equal to 1. In other words, if |r| ≥ 1, the series diverges, and the terms increase without bound.
Example: Let's consider the geometric series with the first term (a) of 3 and a common ratio (r) of 2.
The series will be: 3, 6, 12, 24, 48, ...
The absolute value of the common ratio is |2| = 2, which is greater than or equal to 1. Therefore, this series is divergent.
In a divergent series, there is no finite sum or a limit that the series tends towards. The terms of the series continue to grow indefinitely.
It's important to note that when |r| = 1, the series may either converge or diverge. To determine if it's convergent or divergent in such cases, further analysis or use of alternative methods is required.
In summary, to determine if a geometric series is convergent or divergent, check if the absolute value of the common ratio is less than 1 (convergent) or greater than or equal to 1 (divergent).