Find the MacLaurin for f(x)=x*e^(-x^2)?

Answer 1

Use the MacLaurin series for e^x, it is:

Σ(x^n)/n!

Now, put (-x^2) into it, and get

Σ[(-x^2)^n]/n! = Σ(-1)^n * (x^2n)/n!

To get f(x), multiply the series by x:

f(x) = Σ(-1)^n * x^(2n+1)/n!