Question

- 8) Find the interval of convergence for the following power series: no (n+1)(n+a) no (2n)!" 9) Using f(x) = 8 X = 1 hod a power senes representation for the no 1-X given Anchons (a) f(x) = 2 b) 60) = 1 C) KW) = Orctan (x). I 4- 3x3 +3x² 10) Find the taylor polynomial of degree for the fonction f(x) = V15+x. LÔ 0 - 1 = | b) If n o ano then Ean converges True / False b) If { an is divergent then Elanl is direment: Twe/ False -) If {an 3 is a sequence and i s an 5, then can converses : true False

Answer 1

Page given that fee) => x= n =0 » HX) - 2 4-3x HIV - 2 43.21 Leo 2018 - f(x) = 1 (1+2 x + 3x) + ( ) +----| ho (6) fix) = 1 343x2 Hx) = 1 361 +42) hx) = 1 (1+x²)-1 3 hex) = { [!-x?+(23)2–63)3 +(22)%... thex) = 1 (1-x2 + x4 – x6 +38----| hex) = Ž Į (-1)" (x)22 Recx) = gretain K(x) = taula KIX) = S (-1)*(x)ºn de n nzo O 9 n=0

Date Page = (-1)n xente n-o anti te 2 nt1 n =o 10 WYSO Z Czo *rx) = Z C-119 (De) 2n71 2ntl NA given that f(x) = 1572 To find taylor polynomial of degree 2 about x=1=a. by taylor series- fex) = f(a) + f (a)(x-9 ) + F" (a) (4-9) 2 - - Hence, neglecting higherorder termes, we find a degree polynomial. fix) = feat f'(9)(x-9) + f"(9)(x-4)2 21 Now f(x) = Jista 40 = Jisti = 4 f (x) = -1152 - 2 "(c) = L*64)(1572) W f" (0) = =+116) 12 7" (1) = xloxy = 15 Teacher's Signature.....

Date Page f(x) = 4 + 1 (2-1) - 8 25682 (x-1)? F(x) = 4+1 (at) - 1 (2- 12 Eyla) (b) If lim An=0 then Ian converges - n converges - no talse False If Ean is divergent then Elan absolutely is divergent (false) ter poo Elanl is-abs. convergent If {an} is a segh and lim an = 5 then { an} ( nyo is converge - True.

Date - Page AW @19 Given that the series E ho 2n! let an- 2n) anto = {n+1)! 2n! 1= R lim Hool antil an lina nto I alr I 1 = lim Rento 1 2n+nI L= limit htoontl R 7 R = 0 Thus pauser interval of radius of converter is - 12/<RI | x) / - - olx coo m ebl s e-han no ant) enta) let an = 1 a tanta) anti= (-ight (2+2) (n+3 ) = I'm Wight 2 x Inti) (hta) R 400 (+2)(n+3) - (ion - lim I litt) T (1+3) T no

Date Page LEI R TRELL Hence, interval of radius of convergence is 1x LR a la la 3112X <17 1o E (30) - A=0 (2n)! Met an = (3n) (20) ! duonto forrowed only - there does not exist xn. then how to find internal.