A geometric series is defined as a series of the form arn, where a is the first term and r is the common ratio. To determine if the given series 10, 4, 16, 064 is convergent or divergent, we need to find the common ratio.
Calculating the common ratios between consecutive terms:
- From 10 to 4: r = 4 / 10 = 0.4
- From 4 to 16: r = 16 / 4 = 4
- From 16 to 064: r = 064 / 16 = 4
We see that the common ratio varies: the first ratio is 0.4 and the next two are 4. Therefore, the series does not have a consistent common ratio, indicating that it is not a geometric series in the traditional sense.
To determine convergence, we typically look for a constant ratio in a geometric series. The series can only converge if the absolute value of the common ratio is less than 1 (|r| < 1). Here, since the series does not maintain a consistent common ratio and fluctuates, it diverges.
Thus, we conclude that the series is divergent, and we cannot find a sum for it as it does not converge.