Question

#1.) Find the Taylor series for the following function about the point zo and determine the radius of cowwergence: Cosz → 20 = -2

Answer 1

Solo To find Zo=-T and Taylor series for cosz at radius of convergence 2 Taylor series f(z) = flzo) + F'Es) (z-Zs) + "(70) E-zo)* = --. TOOO71 PTTTTTT 1171749 no Zo =-* f(Z) = COS Z f'(z) = - sinz f"(Z) = - COS Z f"(z) = sinz f"(z) = COSZ f(Z) = f'(Z) = f(8)[Z) = --- - COS Z f'(Z) = A5 (Z) = f() (2) = . =- sinz Fll}(Z) = f(6) (Z) = f(10)(Z) = - = - cos z flll (Z) = f (Z) = f(MZ) = --- = sin Z Costa)=0 & f (za) = f'(zo) = f (4) (Z0) = f (6) 17 - (Zo) --- Scanne2 lith

Now equation (*) can be written as f(Z) = 1. (7+) :-(Z+1) + (2.4%). - + 7 ZAT 2n + Šavi zarz** 2n+1 ZET f(Z) = COS Z = n=o (2ntur ( 22 ) Radim of convergeree (R) of I cn (z-aj” R = lim T Cn h=0 И-9 | Cat . cn = (-)" (2n+i)! 7 lim 1 on 1 = lim (2n + 3)! . hyo cnt no 12ntil = lim (2n+3) (2n+2) ntos Hence = 0 (infinite) Radius of convergence s os. Ha Scanned-with- CamScanner