A geometric series is a series of the form a, ar, ar2, ar3, …, where:
- a is the first term
- r is the common ratio
To determine whether a geometric series converges or diverges, we look at the common ratio r. The series converges if the absolute value of the common ratio is less than 1 (|r| < 1), and it diverges if |r| is greater than or equal to 1 (|r| ≥ 1).
Let’s analyze the given series: 9, 8, 649, 51281.
First, we need to find the common ratio. The common ratio r can be calculated by dividing any term by its preceding term:
r = 8 / 9, r = 649 / 8, r = 51281 / 649.
– For the first ratio: r = 8 / 9 ≈ 0.8889.
– For the second ratio: r = 649 / 8 = 81.125.
– For the third ratio: r = 51281 / 649 = 78.926.
As we can see:
- The first ratio is less than 1 (r ≈ 0.8889).
- The subsequent ratios are greater than 1, indicating that the behavior of this series is not consistent.
Since the common ratio changes and is greater than 1 for the last two calculations, we can conclude that the series diverges. Thus, it does not have a finite sum.
In summary, the geometric series 9, 8, 649, 51281 is divergent because the common ratio is not consistently less than 1 throughout the series. Therefore, we cannot find a convergent sum for this series.