The convergent series are the geometric series with ratios of one-fifth and one-tenth, as their common ratios are between -1 and 1. Other given series are divergent because their terms do not approach zero as n approaches infinity. Options c. and d. are correct answers.
Step-by-step explanation:
When determining whether a series is convergent or divergent, specific tests and properties can be applied to each series presented.
Sigma-summation ∑ from n = 1 to infinity of 2n / (n + 1): As n approaches infinity, this expression simplifies to a ratio of 2, which suggests the series is divergent because the terms do not approach zero.
∑ from n = 1 to infinity of (n² - 1) / (n - 2): Like the previous series, as n approaches infinity, the terms of this series simplify to n and so the series is also divergent for the same reason.
∑ from n = 1 to infinity of (one-fifth)ⁿ: This is a geometric series with a common ratio of one-fifth, which is between -1 and 1. Consequently, this series is convergent.
∑ from n = 1 to infinity of 3 x (1/10)ⁿ: This is another geometric series with a common ratio of one-tenth, which is also between -1 and 1, making this series convergent as well.
∑ from n = 1 to infinity of 1/10 x (3)ⁿ: This series has a common ratio of 3, which is greater than 1, leading to divergence since the terms grow without bound.
The geometric series with the ratios of one-fifth and one-tenth are convergent series. Options c. and d. are correct answers.