Question

match the following series with the series below in which you can compare using the limit comparison test. then determine whether the series converge or diverge. a. $sum_{n = 1}^{infty}\frac{1}{n}$, b. $sum_{n = 1}^{infty}\frac{1}{n^{2}}$, c. $sum_{n = 1}^{infty}\frac{1}{n^{3}}$, and d. $sum_{n = 1}^{infty}\frac{1}{n^{3/2}}$ 1. $sum_{n = 1}^{infty}\frac{1}{sqrt{n^{3}+1}}$ does this series converge or diverge? converges 2. $sum_{n = 3}^{infty}\frac{n + 2}{(n + 1)^{3}}$ does this series converge or diverge? converges 3. $sum_{n = 2}^{infty}\frac{1}{2 + n^{3/2}}$ does this series converge or diverge? diverges 4. $sum_{n = 1}^{infty}\frac{n^{2}-1}{n^{5}+2n^{2}+1}$ does this series converge or diverge? converges

Explanation:

Answer:

(1) D. \(\sum_{n = 1}^{\infty}\frac{1}{n^{3/2}}\), Converges
(2) C. \(\sum_{n = 1}^{\infty}\frac{1}{n^{3}}\), Converges
(3) D. \(\sum_{n = 1}^{\infty}\frac{1}{n^{3/2}}\), Diverges
(4) C. \(\sum_{n = 1}^{\infty}\frac{1}{n^{3}}\), Converges