Polynomials \(a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n\) are nice to evaluate because they rely on addition and multiplication and because we understand them very well. By extension, power series \(a_0 + a_1 x + a_2 x^2 + \cdots\) are also nice, but we need to work to understand them better.

Given any function \(f\), is there a (best) linear approximation to \(f\) at a point \(x=a\)? What about a (best) quadratic, cubic,…, or degree \(n\) polynomial approximation to \(f\) at \(a\)? If we let the degree \(n\) go to infinity, does the power series converge to \(f\) near \(a\)? For what values of \(x\) will the power series converge to \(f\)?