What is the power series representation of the polynomial fu

What is the power series representation of the polynomial function f(x)=5x^3-13x^2+10x-5? What is the radius of convergence? Explain.

Solution

f(x)=5x^3-13x^2+10x-5 itself is the power series representation about x=0.

As f(x) is a polynomial, its power series contains only a finite number of terms known as polynomial.

Since f(x) is a polynomial, its derivative of all orders exists in a neighbourhood of x=0, and therefore its radius of convergence is infinity.