The term "Taylor function" is not a standard mathematical term. The correct concept is "Taylor series" or "Taylor expansion". A Taylor series is a way to represent a function as an infinite sum of terms, where each term is calculated from the function's derivatives at a single point. This allows us to approximate complex functions using polynomials.
The Taylor series formula represents a function as an infinite sum. The general formula is f of x equals the sum from n equals zero to infinity of f to the nth derivative at point a, divided by n factorial, times x minus a to the nth power. This expands to f of a plus f prime of a times x minus a, plus f double prime of a over 2 factorial times x minus a squared, and so on. Each term uses higher-order derivatives and factorial denominators.
Let's work through a concrete example: finding the Taylor series for e to the x centered at x equals zero. For the exponential function, all derivatives are equal to e to the x. When we evaluate at x equals zero, all derivatives equal one. This gives us the series: e to the x equals one plus x plus x squared over two factorial plus x cubed over three factorial, and so on. For example, e to the first power is approximately one plus one plus one half plus one sixth plus one twenty-fourth, which equals about two point seven zero eight, very close to the actual value of e.
泰勒级数是数学中一种强大的工具，它可以将复杂的函数表示为无限多项式的形式。通过使用越来越多的项，我们可以越来越精确地逼近原函数。比如指数函数e的x次方可以用1加x来线性逼近，这就是泰勒级数的第一项和第二项。
泰勒级数的定义是：函数f(x)在点a处的泰勒展开式等于f(a)加上f'(a)乘以(x-a)，再加上f''(a)除以2的阶乘乘以(x-a)的平方，依此类推。用求和符号表示，就是从n等于0到无穷大，f的n阶导数在a点的值除以n的阶乘，再乘以(x-a)的n次方的求和。
这里展示几个常见函数的泰勒级数。指数函数e的x次方等于1加x加x平方除以2的阶乘加x立方除以3的阶乘，依此类推。正弦函数等于x减去x立方除以3的阶乘加上x的5次方除以5的阶乘。余弦函数等于1减去x平方除以2的阶乘加上x的4次方除以4的阶乘。几何级数1除以1减x等于1加x加x平方加x立方，依此类推。
泰勒级数的收敛性展示了随着增加更多项，逼近效果如何改善。级数在其收敛半径内收敛。对于e的x次方，级数在所有实数上都收敛。我们可以看到从只有1加x开始，然后加上二次项，再加上三次项，每增加一项都使红色的逼近曲线更接近蓝色的原函数。这展示了泰勒级数在逼近复杂函数方面的强大能力。
泰勒级数在现代科学技术中有广泛应用。在数值计算中，计算器使用泰勒级数来计算三角函数和指数函数。在物理学中，用于小角度近似和振动分析。在工程学中应用于信号处理和控制系统设计。在计算机科学中用于算法优化和数值方法。在经济学中用于风险分析和期权定价模型。例如，正弦0.1约等于0.1减去0.1的三次方除以6，等于0.09983。泰勒级数是现代科学技术不可缺少的重要数学工具。