Lesson 1

So, we have learned to transition from a sequence to a function. Below, we will introduce a basic tool for the reverse operation—representing a function as a power series, i.e., a Taylor series. In fact, we will be satisfied with its special case, known as the Maclaurin series. The Maclaurin series of a function \(f\) is the power series \[f(0)+\frac{f'(0)}{1!}x+\frac{f^{(2)}(0)}{2!}x^{2}+\frac{f^{(3)}(0)}{3!}x^{3}+\dotsb=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^{n} \ .\]