Someone please help me with the maclaurin series! how would you do this?

Question

Find the Maclaurin series for f(x) using the definition of a
Maclaurin series. [Assume that f has a power series expansion. Do not show that Rsubn (x) -> 0.] Also find the associated radius of convergence.
f(x)=xe^x

Answer 1

f(x) = e^x
e^x = sum(n=0 to infinity)[x^n/n!]
f(x)= x sum(n=0 to infinity)[x^n/n!]
= sum(n=0 to infinity)[x^(n+1)/n!]

(x^(n+2)/(n+1)!)/(x^(n+1)/n!) = x/(n+1) -->0 for all x so the radius of convergence is infinity

Answer 2

Just start differentating. f' = xe^x + e^x. f'' = xe^x + 2e^x. And so on. It's not hard to find a general term for the nth derivative.