Question

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

Answer 1

Refer to explanation

Explanation:

Let `f(x)=e^x` to find series coefficients we must evaluate

`(d^k/dx^k(f(x)))_(x=0)` for `k=0,1,2,3,4,...`

Because `f(x)=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!)`

The radius of convergence is

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