To determine whether the series is convergent or divergent, we first need to identify the type of series we are dealing with. A geometric series has the general form:
a, ar, ar², ar³, …
where a is the first term and r is the common ratio between consecutive terms. For a geometric series to converge, the absolute value of the common ratio must be less than 1, i.e., |r| < 1. If |r| ≥ 1, the series is divergent.
Now, let’s analyze the series provided: 10, 9, 8, 7, 10, 72, 9, 100. If we calculate the ratios of the consecutive terms:
- 9 / 10 = 0.9
- 8 / 9 ≈ 0.888
- 7 / 8 = 0.875
- 10 / 7 ≈ 1.429
- 72 / 10 = 7.2
- 9 / 72 ≈ 0.125
- 100 / 9 ≈ 11.111
Since the series does not maintain a constant ratio (the ratios are not the same for each pair of consecutive terms), it is not a geometric series. In fact, the sequence does not exhibit a clear pattern of convergence. Because the ratios vary widely, and some exceed 1, we conclude that this series is divergent.
In summary, we cannot classify the series 10, 9, 8, 7, 10, 72, 9, 100 as convergent because it does not meet the criteria of a geometric series due to a fluctuating common ratio. Therefore, there is no finite sum to calculate.