To determine whether a geometric series is convergent or divergent, we first need to identify the common ratio (r) between subsequent terms. A geometric series converges if the absolute value of the common ratio is less than 1 (< |r| < 1) and diverges otherwise.
Let’s analyze the series:
- First term (a) = 8
- Second term = 7
- Third term = 498
- Fourth term = 34364
Next, we calculate the common ratios:
- From the first to the second term:
r = 7/8 = 0.875 - From the second to the third term:
r = 498/7 ≈ 71.14 - From the third to the fourth term:
r = 34364/498 ≈ 68.94
As we can see from the values of r:
- The common ratio from the first term to the second is less than 1 (< |0.875| < 1).
- However, from the second term onward, the ratios become significantly greater than 1 (71.14 and 68.94).
In this case, since the common ratio varies and includes values greater than 1, we conclude that the series does not exhibit a consistent convergence behavior. Therefore, we can categorize the overall series as divergent.
In conclusion, the geometric series defined by the terms 8, 7, 498, and 34364 is divergent because the common ratio does not remain consistently less than 1. Instead, we observe values greater than 1 as we progress through the series.