To determine if the given geometric series is convergent or divergent, we first need to identify the common ratio of the series. The series provided is: 10, 8, 64, 512.
A geometric series has a common ratio r that is found by dividing any term by the previous term. Let’s calculate the common ratio:
- From 10 to 8: r = 8 / 10 = 0.8
- From 8 to 64: r = 64 / 8 = 8
- From 64 to 512: r = 512 / 64 = 8
As we can see, the common ratio changes from the first pair (0.8) to the next pairs (8), which indicates that we do not have a consistent common ratio throughout. For a series to be geometric, the ratio must be constant across all terms.
In a geometric series, if the absolute value of the common ratio, |r|, is greater than or equal to 1, the series is divergent. If |r| is less than 1, it is convergent. Since the ratios vary significantly and do not conform to the properties of a geometric series, we can conclude:
This series is not strictly a geometric series and cannot be classified as convergent or divergent in the traditional sense. Therefore, we cannot find a sum for it as we would for a proper geometric series.
If we were examining a geometric series, for instance, if it were 10, 10, 10, 10… (which has a common ratio of 1), we would state that it is divergent. If it were 10, 5, 2.5… (with a common ratio of 0.5), then it would converge, and we could compute the sum using the formula:
- Sum S = a / (1 – r), where a is the first term and r is the common ratio.
In summary, the series provided does not fit the definition of a geometric series sufficiently to determine convergence or divergence, nor can we compute a meaningful sum.