gen-in-3-4-sentences-explain-a-taylor-series-state-any-key-formu-cf0c

In 3-4 sentences, explain a Taylor series. State any key formula.

A Taylor series is a way to represent a function as an infinite sum of polynomial terms, each derived from the function's derivatives at a single point. It allows us to approximate complicated functions using polynomials, which are easier to work with computationally. The key formula for a Taylor series centered at point $a$ is: $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots$$ When $a=0$, this is specifically called a Maclaurin series.