Final answer:
The first geometric series with a common ratio of 0.55 converges with a sum of approximately 44.44. The second series, which seems to be a p-series, also converges due to the p-value of 2. The convergence or sum of the third series cannot be determined without additional context.
Step-by-step explanation:
To determine whether a geometric series converges or diverges, we need to look at the common ratio of the series. For a geometric series with the form ∑ ar^n (where a is the initial term and r is the common ratio), the series converges if |r| < 1, and diverges if |r| ≥ 1. When it converges, the sum of the series can be calculated using the formula S = a / (1 - r).
For the first series, the common ratio is 0.55, which is less than 1. Therefore, the series converges, and its sum can be found by using the above formula:
S = 20 / (1 - 0.55) = 20 / 0.45 ≈ 44.44
The second series is not a geometric series because it does not have a common ratio. The 1/n^2 term suggests a p-series, which converges if p > 1. Since the power here is 2, this series converges, but the sum is not requested.
The third series asks for telescoping sum behavior. To identify this, we must often rewrite or decompose the terms to reveal cancelations within the series. However, without more context or the correct format of the series, it's not possible to provide further details on convergence or the sum.