When calculating without an electronic device for a "small" real number a, many people replace it with the easier calculation of ! Is this shortcut legitimate? The real question should instead be: how much error does it introduce? Can we replace by when x is "close" to 0?
More generally, when x is "small," can the value f(x) of an arbitrary function f be replaced by P(x), where P is a polynomial, making it easier to determine the local behavior of f(x) at 0? And can this be done at values other than 0? How can all this be formalized?
It was not until the late 18th century, when the concept of a limit had become clearer, that the notion of a Taylor expansion was put on a rigorous footing (see "The origins of function expansions"), providing answers to all these questions.
What is a Taylor expansion?
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Let f be a real-valued function defined on a neighborhood of 0 (for example, on an open interval (−a, a), with a > 0), and let n be an integer greater than or equal to 1. The function f has a Taylor expansion of order n at 0 if there is a polynomial P of degree at most n such that, for x in this neighborhood, f(x) – P(x) is negligible compared with *xn: this means that f(x) – P(x) can be written as xn ε(x), where ε(x) tends to 0 as x tends to 0. This is also expressed by saying that f(x) – P(x) is o(xn) (pronounced "little o of xn*").
