Question

Which of the following geometric series converges? I. 2.9 - 3.19 + 3.509 - 3.8599 + ... II. 0.3 + 0.12 + 0.048 + 0.0192 + ... III. 4 - 16 + 64 - 256 + ...

a. I only
b. II only
c. III only
d. I, II, and III

Answer 1

Among the three given geometric series, only Series II converges because its common ratio is 0.4, which meets the convergence criterion of having an absolute value less than one.

Step-by-step explanation:

To determine which geometric series converges, we can look at the common ratio (r) of each series. A geometric series converges if the absolute value of the common ratio is less than one (|r| < 1).

• Series I: 2.9 - 3.19 + 3.509 - ... The common ratio is -1.1 (every subsequent term is multiplied by -1.1). Since |r| > 1, this series does not converge.
• Series II: 0.3 + 0.12 + 0.048 + ... The common ratio here is 0.4. As |r| < 1, this series does converge.
• Series III: 4 - 16 + 64 - ... has a common ratio of -4, which means |r| > 1, so this series does not converge.

Therefore, the correct answer is that only Series II converges.