To determine whether a given series is convergent or divergent, we need to examine the properties of geometric series. A geometric series is typically in the form of:
S = a + ar + ar^2 + ar^3 + …
where a is the first term and r is the common ratio between the consecutive terms. The series converges if the absolute value of the common ratio |r| < 1 and diverges if |r| >= 1.
In this case, the given series is: 8, 7, 498, 34364. Let’s calculate the common ratios:
- From 8 to 7: r1 = 7 / 8
- From 7 to 498: r2 = 498 / 7
- From 498 to 34364: r3 = 34364 / 498
Calculating these values, we get:
- r1 = 0.875
- r2 ≈ 71.14
- r3 ≈ 68.92
As we can see, the values of r2 and r3 are both significantly greater than 1. Since at least one of the ratios is greater than 1, we conclude that the geometric series diverges.
Thus, the series 8, 7, 498, 34364 is Divergent.