# How do you find a power series representation for e^(-x^2) and what is the radius of convergence?

Oct 24, 2015

Use the power series for ${e}^{t}$ and substitution to find:

e^(-x^2) = sum_(n=0)^oo (-1)^n/(n!) x^(2n)

#### Explanation:

e^t = sum_(n=0)^oo t^n/(n!)

Substitute $t = - {x}^{2}$ to find:
e^(-x^2) = sum_(n=0)^oo (-x^2)^n/(n!)=sum_(n=0)^oo (-1)^n/(n!) x^(2n)
Which will converge for any $x \in \mathbb{R}$, so has an infinite radius of convergence.