Question

Exercise 2.14. Determine whether the sequence (sn) is convergent or di- vergent. If convergent, find the limit. Show your reasoning. a) 2n3 -79n2 +42 4n5/2 b) 2n3 -79n2 +42 c) 2n3 79n +46 d) e-cos sin Vn In n Sn- rt 23n Sn 32n (-1)n vn +1

Need to derive proofs for given expressions

Answer 1

"(a) S_n = 2n^3 -79n^2 + 42/4n^5/2 + 1 lim s_n = lim n^3(2-79/n + 42/n^3)/n^5/2 (4 + 1/n^5/2 = lim n^1/2(2 79/n + 42/n^3)/(4 + 1/n^5/2) lim (n^1/2) lim(2- 79/n + 42/n^3)/4 + 1/n^3/2) = infinity (2/4) = infinity lim s_n Rightarrow infinity divergent b) s_n = 2n^3 + 79n^2 + 42/5n^3 + 1 lim s_n = lim 2n^3 -79n^2 + 42/5n^3 + 1 = lim n^3(2-79/n + 4-2/n^2)/n^3(3 + 1/n^3) = lim(2-79/n + 9-2/n^2)/(5 + 1/n^3) = 2-0 + 0/5 + 0 = 2/5 finite then s_n is convergent c) s_n = 2n^3-79n^2 + 46/5n^4 -100n^3 + 1 lim s_n = lim 2n^3 -79n^2 + 46/5n^4 -100n^3 + 1 = lim n^5(2-79/n + 46/n^3)/n^4(5 -100/n + 1/n^4 = lim 1/n(2-79/n + 46/n^3)/(5-100/n + 1/n^4) \r\n lim s_n = lim (1/n) (2 -79/n + 46/n^3)/5-100/n + 1/n^4) 0. (2/3) = 0