Final answer:
For question a), the given series converges to the sum of 2. For question b) and c), the given series diverge because their common ratios' absolute values are greater than 1. For question d), additional tests are needed to determine convergence as it is not a simple geometric series.
Step-by-step explanation:
To determine if each series converges or diverges, and to find the sum if it converges, we can use our knowledge of geometric series. A geometric series is of the form ∑ a r^n, where a is the first term, r is the common ratio, and n starts from 0 or 1, depending on the context.
- a) ∑_{n=1}^{∞} (1/2)^n: This is a geometric series with a = 1/2 and r = 1/2, which is a common ratio less than 1. The series converges and its sum is 1/(1 - r) = 1/(1 - 1/2) = 2.
- b) ∑_{n=1}^{∞} (-3)^n/2^n: This series can be rewritten as ∑ (1/2 * -3/2)^n, revealing a geometric series with a = -3/2 and r = -3/2. Since |r| > 1, the series diverges.
- c) ∑_{n=1}^{∞} 5^n/3^n: This series simplifies to ∑ (5/3)^n, which is a geometric series with a = 5/3 and r = 5/3. Because r > 1, the series diverges.
- d) ∑_{n=1}^{∞} n^2/2^n: Unlike the other series, this is not a simple geometric series due to the presence of n^2. Additional techniques, such as the ratio test or root test, are needed to determine convergence. Without further calculations, however, we cannot conclude the sum or convergence of this series.