Expand (1-3x)e^3x as far as the term in x^4 using a power series for e^x.?

Answer 1

(1 - 3x)e^3x =
(1 - 3x)[1 + 3x + 9x^2/2 + 27x^3/3 + . . . .] =
[1 + 3x + 9x^2/2 + 27x^3/3! + 81x^4/4! . . . .] + [ - 3x - 9x^2 - 27x^3/2 - 81x^4/3! - . . . .] =
1 + 3x(1 - 1) + 9x^2(1/2 - 1) + 27x^3(1/6 - 1/2) + 81x^4(1/24 - 1/6) + . . . =
1 - (9/2)x^2 + 27(-2/6)x^3 + 81(-3/24)x^4 + . . . =
1 - [(1/2)(3x)^2 + (2/3!)(3x)^3 + (3/4!)(3x)^4 + . . .] =
1 - (3x)^2[1/2 + 3x(2/3! + 3x(3/4!))] (calculation form to the x^4 term)

Answer 2

Yeah, yeah, you're a clever little SOB. Just keep yourself away from the boys room, or you'll never get to MIT.