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

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Answer 1 · 2015-10-24T19:40:15

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

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

with infinite radius of convergence.

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

with infinite radius of convergence.

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 $x in RR$ , so has an infinite radius of convergence.

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