Tentukan deret MacLaurin dari f(x) = e^(-x) sin(x)

Tentukan deret MacLaurin dari f(x) = e^(-x) sinx

Jawab:

Deret MacLaurin;
f(x) = f(0) + f^1 (0)(x)+(f^2 (0))/2! x^2+(f^3 (0))/3! x^3+(f^4 (0))/4! x^4+⋯

Maka:
f(x) = e^(-x) sinx
f^1 (x)= -e^(-x) sinx+e^(-x) cosx
f^2 (x)= e^(-x) sinx – e^(-x) cosx  -e^(-x) cos⁡x – e^(-x) sinx =-2e^(-x) cosx
f^3 (x)= 2e^(-x) cos⁡ (x) + 2e^(-x) sin⁡x
f^4 (x)= -2e^(-x) cos⁡ x -2e^(-x) sin⁡x-2e^(-x) sin⁡ x+2e^(-x) cos⁡x

Maka:
f(0) = e^(-0) sin⁡ 0=0
f^1 (0)= -e^(-0) sin⁡ 0 + e^(-0) cos⁡0=0+1=1
f^2 (0)=-2e^(-0) cos⁡ 0 = -2
f^3 (0)= 2e^(-0) cos⁡ 0+2e^(-0) sin⁡ 0=2+0=2
f^4 (0)= -2e^(-0) cos⁡ 0-2e^(-0) sin⁡ 0 -2e^(-0) sin⁡0+2e^(-0)cos⁡0 = -2-0-0+2=0

Maka: deret macLaurinnya

f(x) = f(0) + f^1 (0)(x)+(f^2 (0))/2! x^2+(f^3 (0))/3! x^3+(f^4 (0))/4! x^4+⋯
f(x) = 0 + (1)(x)+(-2)/2! x^2+2/3! x^3+0/4! x^4+⋯
f(x) = 0 + x-x^2+1/3 x^3+0+⋯
f(x) = x-x^2+1/3 x^3+⋯