Question

n=0 4. Using the power series cos(x) = { (-1)",2 (-0<x<0), to find a power (2n)! series for the function f(x) = sin(x) sin(3x) and its interval of convergence. 23 Find the power series representation for the function f(2) and its interval (3x - 2) of convergence. 5. +

Answer 1

A13.4 Given Cosby = 2 (-) 220 2m! mco fu Sinx) sin(3x) f(x) तान [2 Six) Sh(3N)] %3D 2017 । Cas (x - 34) – CoS (+3x3] %3D f(x) 1 1 2 [ Cos(2) Cos (५४)] fla) 1 Cosex) Cos(५४) 2 flow IN तान 2 (-1) (43) 21 (i)" 92) (On), nco (On)! 2m fox (A) (2) 2m. 2n mco 2 तान Mzo £ (4)" (4)" hem On]

flan والے neo (+) (2) 2n an! (4) (4, 2n any (2) 27 This the hower Beries of fla). Now delermined Let invervalo of convergence 2 (-) (22) an! Hod an ban (2) 2n = (226 an! So f(x) MED n NELO find Radius of Convergence for both series. lim - = Ant Ri 1 R, lim dlox (-) + (2, 2n+2 (x 2n+2 х (2n+2)! 2.2n! (1) (2) 2n (2120 11 lim (2². (2) 2 2n+2) (2n+1) 0 1 R, Hence R=0 Hence intervalore Radius of of convergence Conv. of is infinite. So is R.

Now find Radius of Convergence for bn 1 R2 lim briti no bn 1 R₂ lim 61) + (4) 2n+2 6u² (2n+2)! (2) 2n+2 q x Z. (2n)! (4) (4,20 (xan = lim (4) 2 (2) ² (2n+2) (2n+1) 1 R₂ O 11 R₂=00 5 So Infinite. Hence interve Radius of convergence of convergence is R. Since Buth the Series have interval of conv. is of of R. Hence Interval cons- flu is R ie (-o