Question

02. Investigate the convergence or divergence of the following series. State the name of the test you used n2 +1 n2 +5n +2 9n3-2 nel

Answer 1

Cauchy's nth root Test let Sigma a_n be a positive term series such that Lim_n rightarrow infinity (a_n)^1/n = l then i) Sigma a_n converges if l, 1 (ii) Sigma a_n diverges if l > 1 iii) test fail if l = 1 9a) Sigma_n = 1^infinity (5n + 3n^3/9n^3 - 2)^n a_n = (5n = 3n^3/9n^3 - 2)^n Lim_n rightarrow infinity (a_n)^1/n = Lim_n rightarrow infinity (5n + 3n^3/9n^3 - 2)^n/n - lim_n rightarrow infinity (5n + 3n^3/9n^3 - 2) = Lim_n rightarrow infinity n^3 (5/n^2 + 3/9 - 2/n^3) = 3 + 0/9 - 0 = 3/9 = 1/3, 1 by Cauchy's nth root test series is converges \r\n (b) Sigma_n = 1^infinity tan^-1 (n^2 + 1/n^2 + 5n + 2) a_n = tan^-2 (n^2 + 1/n^2 + 5n + 2) Lim_n rightarrow infinity a_n = lim_n rightarrow infinity tan^-1 (n^2 + 1/n^2 + 5n + 2) = tan^-1 (lim_n rightarrow infinity (n^2 + 1/n^2 + 5n + 2))