Calculus 2 , series

Question

Find the maclaurin series for the function :

f(x) = x^2 . e^ -3x

Answer 1

The Maclaurin Series for e^x is 1 + x + x²/2! + x³/3! + x⁴/4! + ...

So, the Maclaurin Series for e^-3x is 1 + (-3x) + (-3x)²/2! + (-3x)³/3! + ...

Therefore, f(x) = x²e^-3x = x²(1 - 3x + 9x²/2! - 27x³/3! + ...) = x² - 3x³ + 3²x⁴ /2! - 3³x⁵/3! + ...

= ∑_{(n=0 to infinity)} [(-3)ⁿxⁿ⁺² / n!]