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

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Answer 1 · 2015-09-15T03:29:30

Refer to 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#