在数学中，泰勒级数（英语：Taylor series）用无限项连加式——级数来表示一个函数，这些相加的项由函数在某一点的导数求得。泰勒级数是以于1715年发表了泰勒公式的英国数学家布鲁克·泰勒（Sir Brook Taylor）来命名的。通过函数在自变量零点的导数求得的泰勒级数又叫做迈克劳林级数，以苏格兰数学家科林·麦克劳林的名字命名。

拉格朗日在1797年之前，最先提出带有余项的现在形式的泰勒定理。实际应用中，泰勒级数需要截断，只取有限项，可以用泰勒定理估算这种近似的误差。一个函数的有限项的泰勒级数叫做泰勒多项式。一个函数的泰勒级数是其泰勒多项式的极限（如果存在极限）。即使泰勒级数在每点都收敛，函数与其泰勒级数也可能不相等。开区间（或复平面开片）上，与自身泰勒级数相等的函数称为解析函数。

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

The concept of a Taylor series was formulated by the Scottish mathematician James Gregory and formally introduced by the English mathematician Brook Taylor in 1715. If the Taylor series is centered at zero, then that series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.