Taylor Series

If f(x) is a well-behaved* function near a chosen expansion point, x₀, then the function can be expanded in a series called a Taylor series:

如果函数 f(x) 在选定的展开点 x₀ 附近是良好行为的，那么该函数可以展开为一个称为泰勒级数的级数：

The Taylor series for a function is often useful in physical situations to approximate the value of the function near the expansion point x₀. It may be evaluated term-by-term in terms of the derivatives of the function. It is typical for successive terms in a series approximation of a function to get smaller and smaller, and when the size of the next term appears negligible in terms of the problem being addressed, then one can judge that a sufficiently accurate approximation of the function has been achieved.

函数的泰勒级数在物理问题中常用于近似函数在展开点x₀附近的值。该级数可以逐项用函数的导数来计算。通常，函数的级数近似中后续项会越来越小，当下一项的大小在所研究的问题中可以忽略不计时，就可以判断已获得足够准确的函数近似。

It is often the case that a convenient expansion point is x₀ = 0, and series about this special expansion point are also called Maclaurin series. There are many applications for expansions of common functions around x=0. Some examples are:

通常情况下，选择x₀=0作为展开点会很方便，以这个特殊展开点展开的级数也称为麦克劳林级数。许多常见函数在x=0处的展开有广泛的应用。一些例子包括：

三角函数级数的单项近似有诸多应用，称为'小角近似'。

* Well-behaved means that both the function and its derivatives are defined and continuous in the range of the expansion around x₀.

良好行为意味着在围绕x₀展开的范围内，该函数及其导数都是定义良好的且连续的。