Power series $e^{-x^2}$

Question

How would I create a power series of $f(x)=e^{-x^2}$ around $x_0=1$ without using a Taylor series?

I need to know this for my upcoming exam so I would be really grateful to anyone who could show me how it's done.

Thanks in advance

Answer 1

$$e^{-x^2}=e^{-(x-1+1)^2}=\sum_{n=0}^\infty\frac{(-1)^n(x-1+1)^{2n}}{n!}=\sum_{n=0}^\infty\frac{(-1)^n\sum_{k=0}^\infty{2n\choose k}(x-1)^k}{n!}=\sum_{n=0}^\infty\frac{(-1)^n{2n\choose 0}}{n!}+\sum_{n=0}^\infty\frac{(-1)^n{2n\choose 1}(x-1)}{n!}+\sum_{n=0}^\infty\frac{(-1)^n{2n\choose 2}(x-1)^2}{n!}+\cdots$$