9. Which of the following statements about convergence of the series 30 (A) Σ h=] (Β) 1 In(n + 1) 1 n=t In (n + 1) 1 (C) Σ n=j In(n + 1) 1 (D) Σ n=j In(n + 1) · converges by comparison with ΣΕ n=1 1 · converges by comparison with Σ n=1h- diverges by comparison with Σ h=1 diverges by comparison with Σ - 1 n={h= n=j In(n + 1)
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- 8. Determine the radius of convergence and interval of convergence for the power series n(-1)" 4n n=1 -(x + 3)".Which one of the following sequence or series converges 1-3...5...(2k–1) 10k k²+3k-5 (D) Ek=2 k(ln k)² 1 (A) (B) Σ-0- 1)". (C) En=1 3k2–2k–1 =1 3k²-2k-1 (E) none of (A) - (D)Find the interval of convergence for each power series. | 19. En=1 (-1)n+1xn n2n 2n+1(n+1)(x-1)" 20. Ση-1 3n(n+2)
- Suppose that the series ax has radius of convergence 5 and (a) (b) 2 (an+b)xn n=1 n=1 (c) bn(x - 2)4n n=1 n=1 R = an(x-7) R = R = n=1 box has radius of convergence 9. Find the radius of convergence of the following series.Which of the following statement(s) is/are true? converges conditionally. 00 «Σ(-1)", n=1 (ii) 00 n= n+9 sin(n) Sn² A) (ii) 80 2n+1 (1) Σ (-1) " diverges. n=1 3n+4 (B) (ii), (iii) (C) (i), (ii) converges absolutely. All of them (E) (i), (iii) F None of themSuppose that the power series Σan(x-6) n=1 converges at x = 9. At which of the following points MUST it also converge? (i) x = 8 (ii) x = 6 (iii) x = 3 (A) none of them (B) all of them (C) (i) and (iii) only (D) (i) and (ii) only (E) (ii) only (F) (i) only (G) (ii) and (iii) only (H) (iii) only
- Q5/ For the following series :- 1 (x-2), (x - 2)2 (x- 2)" f(x). + (-1)". - 4 8 2n+1 Find :- a- Values of (x) that make series converged b- The sum of series Ans./ a) 0Determine the convergence or divergence of the p-series 1 + 1/sqrt(2) + 1/sqrt(3) + 1/sqrt(4) + ...Use the Root Test to determine the convergence or divergence of the given series or state that the Root Test is inconclusive. -n2 1 9n n=1 L= lim Vlan| (Enter 'inf for o.) -n² is: E4 (1+ ) O A. convergent O B. divergent OC. The Root Test is inconclusive6. (a) Show that sinh (r) = cosh(2.r) – 1 2 (b) Find the nth term of the following sequence {1,8, 10, 26, 46, -..} for all n 2 1. (c) Define both absolute convergence and conditional convergence of a series. 7. (a) Given a,+1 = 2a, + an-1 and b+1 = 2b, + bn-1 for n 2 1, with ao = a1 = 1 and bo = 0, bị = 1. It can be shown that (1+ v2)" = an + b, V2, Vn 2 0. Hence, prove %3D !! that a - b = (-1)". 2 - VA-x (b) i) Compute lim ii) Hence prove the limit in i) above by e-d definition.9. Estimate the value of the convergent series k=1 (-1)kk² k! + 2k with an absolute error less than 10-5.Which of the following series are conditionally convergent? arctan n (i) 2(-1)"- n? n=1 00 (ii) E(-1)" -1 n Vn* + 5 n=1 (iii) E-1)*+1 cos(1/n) n=2 A) (iii) only (B) (i) and (ii) only (C) (ii) only (D) (i) and (iii) only (E) (ii) and (iii) only (F) all of them G) none of them (H) (i) onlySEE MORE QUESTIONS