Maclaurin Series

Exp Find Maclaurin Series for MS = TS at x=0 MS I f(x) = COSX (-v) is (1) (4) (n) 0 f(0) 2 flo) 3 o f(0) n = f(o) + f(o) X + X + X EN + X X 2! 3! o 4! n! (1) n=0 f(x)= = COSX flo)=1 f = - - sinx flo)=0 " III "f = - cosx f(o)=- f = sin x F101=0 (a) (5) (5) (4) f = cos x flo)=1 f = - sinx f (o)=o sinx o - cost -1 - sins 6 cost I $ 2 x to - 1x + O + = \ to 2! 6 y 2 X X + + - 21 y ! 6! f(x) = cosx VUL Approximation Poly. of degree O Po(x) = I 2 x P2(x) = 1- 2! y =4 2 X 1, PP(X) = 1- x + 4 & n 6 ? X x y S (-1) X

x 0 x y = Cosx - = 2 y 6! 2 (-1) X 1- + (2n)! n=o converges MS n on x x6 Poly of degreen Pn(x)= 1- + + 2: Y! 61 (2n)! lim Pa(x) = Cosx about O n- 8 \ 2 f(x) = sinx MS= TS at O / 10 f flo)=1 f = sin x f(o)=0 = cost " 111 11 f= -sinx "f(6)=0 f = - cost ()=-1 (5) 15) (4) = cost f (01 = 1 (4) f = sint f(a)=6 f : (4) o " (n) f(o) f(o) 3 (10)x4 EN f10) X n = ((o) + f(o) x + X+ 2! 3 4! 57 n! 3 = O t O x O + X - X n=0 + 5! 7. 3 00 n (2n+1) 3 5 7 (11 X X X sin x = X - t till = 31 51 7! (2n+1)! x

sinx|-/|^ 3! 51 71 ( (2n+1)! MS n=0 P,(x) = x x3 vv P3(x) = X- at x=0 31, Ps(x) = x x- + vve 5 3 f(x)= Ms=Ts at x=0 x f= f(o)=1 X f=e f(a) "= =) "f(d)=1 III : 8 (a) X = f(6)++f(a)x + + flo) x x e - = n! x3 + n=0 00 n X x x3 xx C + x + 21 + = 3! n! n=0 conv. MS

Lonu MS X MS e, sinx, (osx x e = + 31 n=d y (-y x Cost = 1- + 2 6! (2n) n=0 3 5 7 n 2n+1 Sinx = X- - + X + (1) 3' 5! 7 = (2n+1)! Exp Find Talyor Series of ((x) = at x = ) a=1 iN f(1) (n) (x- 1) n = f(1)+f(1)(x-\) + fl1) n! n=0 f11) = 2 f=? f = 2 (1n2) 'II) 2 Inz 2 f = 2 X (Ini) "F(1) 2 (1nz) "f = X 2 Unaz and = X

(n) (1n7) =2 +(21n2)(x-1) + 2(1n3) 2! a-1)2 2 (mm) (x-1)-+... D = E 2(122) (+-' 1) n n! n= Exp Find for coshx f(x) X20 MS = X -X [ X _X coshx = e e e = 1/2 = 2 = 1/2 [it x+2' 21 31 + TH Y! + (1+(-x)+ 2! 3! =1/20 (DX+ + Ohr I 6 XX@ "X" t 2 6! 2 = 1 + & 2 + x 41 4! + 6T Exp Find the first 4 nonzero terms in the MS of

Exp Find the first 4 nonzero terms in the MS of D 1/3 (2x + X (osx) 2 X + XIM COSX = 3 6 X x + ....] = MD 2 X + y 61 3 7 3 5 + 3 (4!) 3 (61)+ - }x+1 3(2!) = 5 7 3 X X = X X - + 3(21) 3 (4!) 3 (61) 5 7 X X The 1st y non zero terms : X, 3(21) / (1!) (61) X 2 e sinx 3 5 X X X = 14xtz X + + +4/---- 3 ! .) 31 5! 6 6 2 3 X X X X (31)131) 5! 5! + = 2 31 (2!) 4:

I 2 3 = x x x X + 3! ( x + + 5! X ) + y! 3! (21) 2 COSX = 3 COS2X { (1) (2n)! n=0 2n 6 (N (-v)(2x) X = = 2 6! tim (2n)! 41 16 n=o 4 x2 6 2X (2x) (2x) I + = 2 41 6! f(x)= f10)=1 f = -2sin2x " III "F =-4 COSZX fol= f = 8sinzx (4) (5) 17 = 16 C032X =) f(o)=16 f = - 32 Sin2r i