Question

Please help me. these go together. if you help then i will definitely rate!:)

(a) Use the power series for 1 to prove that the Taylor series centered at x = 0 for In(1+x) is 1+1 + (-1)" 2"41 2 3 4 5 7+1

Answer 1

Taylor series of a function fly about a point 120 is given by - fony = frø+ x f'0+ 2.4f"20+ (0)+-- 33. f"0)+ 14 fly -0 14 (@criven that fow = lnctus Now, f(0) = incito) = 0 f'(x) = te f'(o)=1 HU f(x) = f(0) = -1 (1+r)2 fill (1) fill (o)= 2 citari 1 (173 fay= f(0) = -6 (1 true 1 putting the value of foo, f'(o), f'(o), f'(o) and fly in equation , we get f(x) = fo of ut Som (-1+ 0+ (0+ - f(x) = 21-2² we 32 + x3 00 flug = { (in ahal h0 ht
page1
hoo fou = Incitu) = {(Wh whol (6 (hom How to find interval of convergence but flow = ca ah be a power series Then ite rasting of cowayenu=Rctive licensen h=0 The Interval of convergence in CR,R) Now, R= dim ho () (hr) (1) hot (htt) 00h (h+2) R = elm h3al(htt) (1h R = 1 Thus, interval of convergence is (1,1) Let gCM) = (x-3)? in (1+ **) 거야 713 3 t. + 45 we know that In CI+H) = 4 1-22 + Now . g cry can be written as gone creº (69-1933+0+*+014) gou - 1378 aspectes
page2
glu) = { (ish (21-3ght hao (h+1) 5ht This is the required function gery in taylor series Radius and interval of convergence for the taylor series at 11=3 for goy (h * R = lim hə oo (hta) 5thta) (4) hd (ht) shu R = lim 5 (h+2) (hor) Х hgoo IR=5 Region of Internal of convergence 14-3/<R re 14-3125 = -2<4k8 Interval of convergence is (-2,8)
page3
Here, 8th degree tay for polynomial exparsion at U23 for goy 901) E (4)h (91-3) h+3 (ht) 5ht ho 4 gong = (1-33 (1-3) گز{ -)1 + 2x52 5 55 3.52 ++ (11-336 54 + √(1-3)? ++ (2-338 56 at u=3 This is the required gith degree taylor polynomicel Now g'(3) = 17 56 to g?(3) 5040 15625
page4