Originally Posted by

**Stev381** Hi to you,

Q: Find the Maclaurin series of $\displaystyle f(x) = xe^x $.

Also find the associated radius of convergence.

A: So far I have found this __for the first part__:

$\displaystyle f^n(x) = e^x(n+x)$.

so it follows that:

$\displaystyle f^n(0) = n $.

and so:

$\displaystyle xe^x = \sum\frac{x^n}{(n-1)!}$ n going from 0 to infinity. n = 1 to infinity

That is:

$\displaystyle xe^x = x+x^2+\frac{x^3}{2!}+\frac{x^4}{3!}+....$ fine

I am right for the first part?

__Second part:__

I did the ratio test and I got that the limit is smaller then 1. yes, 0 is < 1

Therefore the sum is absolutely convergent...??? correct