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

Refer to explanation

#### Explanation:

Let $f \left(x\right) = {e}^{x}$ to find series coefficients we must evaluate

${\left({d}^{k} / {\mathrm{dx}}^{k} \left(f \left(x\right)\right)\right)}_{x = 0}$ for $k = 0 , 1 , 2 , 3 , 4 , \ldots$

Because $f \left(x\right) = {e}^{x}$ all coefficients are equal to 1

The power series is

Σ_0^oo(f^(k)(0)/(k!))(x-0)^k=x^0/(0!)+x^1/(1!)+x^2/(2!)+x^3/(3!)+...=Σ_0^oo x^k/(k!)

lim_(k->oo)abs((x^(k+1)/(k+1)!)/(x^k/(k!)))=0