Final answer:
The geometric series with terms 5, -10, 20, -40, etc., diverges because the common ratio is -2, which has an absolute value greater than 1. Option number C is correct.
Step-by-step explanation:
The series provided is a geometric series which can be written in the form 5 - 10 + 20 - 40 + .... To determine convergence, we need to calculate the common ratio (r) by dividing any term by the previous term. In this case, r = -10 / 5 = -2. A geometric series converges if the absolute value of the common ratio is less than 1 (|r| < 1). Since our common ratio is -2, the absolute value of which is greater than 1, the series diverges.
If a geometric series does converge, the sum can be calculated using the formula S = a / (1 - r), where a is the first term of the series, and r is the common ratio. However, since our series diverges, we cannot apply this formula to find its sum.