4. If EC,4" is convergent, does it follow that the following series are convergent? η Ο...
Find the MacLaurin series for the function f(x) = 12, ο ΣΩ(+ 1): ο ο Σ(-1) 0η Σε ο 2) η -0
(9 marks) Using any methods determine whether the given series is convergent. Give a full explanation. a. Σ=1 12n +50m 1 1 b. Σ=4 χ (1x)8 C. Σε η(n+4)(3n2-1) (2n+3)(5n+7) (6n2-1) (n+1)
Determine whether the series is absolutely convergent, conditionally convergent or divergent. 2"m! (b) Σ(-1)". 5 • 8 • 11 •• (3η + 2) (c) Στ (1 + Ae η =1 1 (- 2)" (-1)" (e) Σ (- 1)"e" (f) Σ (g) Σ (n + 1)! η 1 η 2 mln (2017)
2. (a) Determine whether each of the following series converges or diverges. If convergent, find the sum of the series. Σ ο Σβ-) 3. (ii) EL(** + 2) – In (2e* + 3)] (ii) (i) 16k2 + 8k – 3 24 k-1 (b) Give a non-trivial example of two divergent series such that their sum is convergent.
Use power series operations to find the Taylor series at x = 0 for the given function. f(x) = x2 In(1 + 7x) Ο Σ (-1)γιχη+2 η + 1 ΠΟ Ο Σ (-12-17ηχη+2 η + 1 η-Ο Ο Σ (-1η-12η-1_n-1 11 Ο Σ (-11-1γη,0+2 η1 ο η O Ση
7) Use the Ordinary Comparison Test to determine whether the series is convergent or divergent. Υ n (a) (6) Σ η η 5" 3η – 4 M8 M8 (Inn) 2 (c) η (d) tan n2 n3 η-2 1 (e) Σ (6) Σ 2n + 3 2n + 3 ή-1 1-1
Which is series divergent? ο Σ=1 1,000,000+η 1 1 Σ=1 1.2 1 Σ En=l n2 Σ=1 an+1
(a) Starting with the geometric series X?, find the sum of the series η ΕΟ Σ ηχο – 1, 1x] <1. ΠΕ 1 (b) Find the sum of each of the following series. DO Σηχή, 1x <1 η = 1 η (i) Σ. (c) Find the sum of each of the following series. D) Σπίη – 1)x, Ix <1 ΠΕ 2 (i) Σ - η 57 ΠΕ 2 0 i) 22 = 1
Find the sum of each convergent series. Use scratch paper and put your answer in the corresponding blank to the left of each problem. Evaluate and simplify all answers. 4 (a) και m=3 η 4 (b) k=1 (c) 1 1 + 1 4. 24 +... 1. 2 2. 22 3. 2 12 n=0 1 (d) Σ (-5)" (e) Σ (n-3) (f) Σ (2n+1)! n=4 (-1)" π2n+1 2n+1 η=0 (9) 744 * 10 (h) ΣΑ) (3) 1 - In 2 +...
2-15 Determine whether the series is convergent or divergent. 1 2. Σ 1.0001 3. Σ 1-0.00 n=5 η n=1 σο 2 3 4. Σ 5. Σ (1) ده است + ηψη 3 n=1 1 1 6. Σ 7. Σ η=5 (η – 4)? 2n + 3 n=1