I have been talking to a few people about when it is ok to use the Maclaurin series instead of using the Taylor series.
I.e.
Let $f(x)=e^{x^2-2x+1}$.
Write down the degree 3 Taylor polynomial for $f(x)$ centered at 1.
Obviously you can use that $p(x)=f(1)+f'(1)(x-1)+f''(1)(x-1)^2/2+f'''(1)(x-1)^3/3!$, but you can also do $e^{(x-1)^2}=\sum(x-1)^{2n}/n!\approx 1+(x-1)^2$.
Why are you able to use the Maclaurin series instead? Given that the series is supposed to be centered at 1? The Maclaurin series is centered at 0. I know that the parabola $y=(x-1)^2$ is centered at 1. So instinctively I think that this is the reason why. But I would like a more detailed understanding of why this is the case.
EDIT:
To go into a little more detail with what the students were asking. With this example, say that instead of it factoring into $e^{(x-1)^2}$ the function was $e^{(x-2)^2}$ and still had it centered at 1, would I be able to use that same idea of using the Maclaurin to answer the question.
Best Answer
Let $f(x)=\sum_{k=0}^\infty a_k x^k$ be a function with the given Taylor expansion centered at $0$. Then $f(x-x_0)=\sum_{k=0}^\infty a_k(x-x_0)^k$ is a function with the given Taylor expansion centered at $x_0$.
Now apply this to $f(x)=\exp(x^2)$ and $x_0=1$.