Question

(a) Use differentiation to find a power series representation for f(x) = (7+x)2 (-1)"(n+1)x" 72+2 What is the radius of convergence, R? R = 7 (b) Use part (a) to find a power series for - 17+)3 (-1) (n+2)(n+1)x" 2.71+3 n = 0 What is the radius of convergence, R? R = 7 (c) Use part (b) to find a power series for Rx) = x 2 413 As - O What is the radius of convergence, R? R = 7

Answer 1

(9) 400) = GB02 $60) = CH Hole = Differentiate with respect to be *x) = -2 10 , + (0) = -23 *(*= $0) = #Cent) = 2 + (0) = 120 29 By maclawrin scries. 300-asje = $(0) + $04 + shox + f"co) set.. ** 33 -24 31 40) = (as mana"] nta neo

Radius of convergenal of Power sories an= (-1)^(n+1) 7 +2 ² = dim Panti no = les GACH20 1 I 7.8tlo) | (1+2/2) = (to) 7. (1+/0) ACHO) R = 7 (6) 4)= CHUB : +6)= $ca) = Cita f'(o)= - f"() = + 2 , "CO) = {"(x) = -beton , t (o) = -belo {"(x) = 2 r "(O)= 340

BY maclaurin series SG)- TN - N - - + 34 x° +----- - - - - - 2.76 f) = (-15. (en42) (+1). som] - O 2. 7743 Radius of convergenct:- An= (-150 (+2) (n+1) 2.7M+3 +2=lim (-1) + ((n +3) (12) - no 1 z. 2.7nty (-1) (0+2)(n+1) 2. 79+3 =lim م 7 *(1+34) (1+2 ) (1+2m) (tkr) - (1+3% (1+2/0) (1+2 )(1+/6).7 R = R=7

(c) f(x)= if(o)=0 to13 tx) - 2 - 1 +(o)=0 +'@) = (2+0) 22 =(771)3 7 TGV12 = (2+0) [(71)28-308] G+ seja = (1484281²_302) (778)4 if (() = (148-522) (713c34 , ť (o)=0 so = (772)4 (14-2X)-(1472) 4 (772)3 [(Asc) 973 - (+x} [(+2) (14-22) – 4 (147-32)] (7x) 98–1434 +1432-27?– 56+4x? (7505 = f (x)= 98-568+481² C , t (0) = To f"(x) = (2+x)= (-56 +8X) - (98-568+4x²) 5 (77) (6273035] 71004 1-392+567-56x +85² _ 490 +280x-2012) (4x) -882 +2808-1272 (7+2) = 97

BY Maclawin svies. 56) L= = 0+04+ -232 4*+-- = 4 1 . x _ 147 x +- - ... f(x) = { (+59CM+5)(3+12) 8 n=2 2. 2013 Radius of convergenal an= (-1)? (245) (+12) 2. 71+3 - lim | (+) at con+6) +13) no २. +4 An (+5) (+12) 2. 99+3 - din m² (+6M) (+13) - na 7. (1+5lm) (1+122) ☆ R=7