Question

I was given three possible expansion points [x0=1,-1,0] and was asked to determine which can be used to create a valid Taylor expansion of f(x)=xe^x any ideas?

Answer 1

You can prove by induction that the kth derivative of xe^x is\[f^{(k)}(x)=(k+1)e^x+xe^x\]so your Taylor series expansion about a point x_0 would be,\[\sum_{k=0}^{\infty}\frac{(k+1)e^{x_0}+x_0e^{x_0}}{k!}(x-x_0)^k\]Now, technically, expanding about any of those points is 'okay', but I'm thinking, since you're expanding for an exponential, and since we use these expansions to express functions in terms of powers of x, it may defeat the purpose if you select 1 or -1, since you'd have e or 1/e in your expansion (from the kth derivative).

On the other hand, if you select x_0=0, then your exponential terms in the kth derivative term disappear, and you have something more befitting of the purpose of a Taylor series expansion.

Hope this helps.