Find the Taylor series centered at x = -1 for the function

Question

f(x) = x(e^x)

Answer 1

f(x) = x e^x = -1/e
f'(x) = (x+1) e^x = 0/e
f"(x) = (x+2) e^x = 1/e
f³(x) = (x+3) e^x = 2/e
...
x e^x = -1/e + (x+1)^2/(2! e) + (x+1)^3/(3!*2e) + (x+1)^4/(3!*4e) ...
= 1/e (-1 + (x^1)^2/2 + (x+1)^3/3 + (x+1)^4/8 + ...)