taylor's series

简明释义

泰勒级数

英英释义

例句

1.In calculus, we often use taylor's series 泰勒级数 to approximate functions around a certain point.

在微积分中，我们常常使用taylor's series 泰勒级数 来在某个点附近逼近函数。

2.To understand the behavior of polynomials, students learn about the taylor's series 泰勒级数 in their coursework.

为了理解多项式的行为，学生们在课程中学习taylor's series 泰勒级数。

3.The taylor's series 泰勒级数 for sin(x) helps us calculate the sine of small angles easily.

sin(x) 的taylor's series 泰勒级数 帮助我们轻松计算小角度的正弦值。

4.Engineers often rely on the taylor's series 泰勒级数 to simplify complex equations in their calculations.

工程师们常常依赖taylor's series 泰勒级数 来简化计算中的复杂方程。

5.The taylor's series 泰勒级数 expansion of e^x is a fundamental concept in mathematics.

e^x 的taylor's series 泰勒级数 展开是数学中的一个基本概念。

作文

In the world of mathematics, the concept of approximation plays a crucial role in simplifying complex functions and making them more manageable. One of the most powerful tools for this purpose is the Taylor's series, a mathematical representation that expresses a function as an infinite sum of terms calculated from the values of its derivatives at a single point. This series is named after the mathematician Brook Taylor, who introduced it in the early 18th century. The Taylor's series allows us to approximate functions that are otherwise difficult to compute, particularly in calculus and numerical analysis. To understand the significance of the Taylor's series, consider the function f(x) = e^x, which represents the exponential function. By using the Taylor's series centered at x = 0, also known as the Maclaurin series, we can express e^x as:f(x) = e^x = 1 + x + (x^2)/2! + (x^3)/3! + ... + (x^n)/n! + ...This infinite series converges to the true value of e^x for all real numbers x. The beauty of the Taylor's series lies in its ability to provide polynomial approximations of functions, which can be easily computed and analyzed.Moreover, the Taylor's series is not limited to the exponential function. It can be applied to other functions such as sine and cosine. For instance, the Taylor's series for sin(x) centered at x = 0 is given by:sin(x) = x - (x^3)/3! + (x^5)/5! - (x^7)/7! + ...Similarly, the Taylor's series for cos(x) is:cos(x) = 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + ...These series allow us to compute trigonometric functions with great accuracy, especially when using only a few terms of the series. In practical applications, the Taylor's series is widely used in physics, engineering, and computer science, where approximating functions is essential for solving real-world problems.However, it is important to note that the Taylor's series has its limitations. The convergence of the series depends on the function and the point around which it is expanded. For some functions, the Taylor's series may converge only within a certain interval, while for others, it may diverge entirely. Therefore, understanding the radius of convergence is crucial when applying the Taylor's series.In conclusion, the Taylor's series is an indispensable tool in mathematics that facilitates the approximation of complex functions. Its ability to express functions as an infinite sum of polynomial terms opens up numerous possibilities for analysis and computation. As we continue to explore the depths of mathematical analysis, the Taylor's series remains a fundamental concept that connects various fields of study, enabling us to tackle complex problems with greater ease and precision.

在数学的世界中，近似的概念在简化复杂函数和使其更易于处理方面起着至关重要的作用。为此，最强大的工具之一是Taylor's series，这是一种数学表示法，它将函数表示为从单一点的导数值计算出的无限项之和。这个系列以数学家布鲁克·泰勒的名字命名，他在18世纪初引入了这一概念。Taylor's series使我们能够近似那些难以计算的函数，特别是在微积分和数值分析中。为了理解Taylor's series的重要性，考虑函数f(x) = e^x，它表示指数函数。通过使用以x = 0为中心的Taylor's series，也称为麦克劳林级数，我们可以将e^x表示为：f(x) = e^x = 1 + x + (x^2)/2! + (x^3)/3! + ... + (x^n)/n! + ...这个无限级数对所有实数x收敛到e^x的真实值。Taylor's series的美在于它能够提供函数的多项式近似，这些近似可以轻松计算和分析。此外，Taylor's series不仅限于指数函数。它还可以应用于其他函数，如正弦和余弦。例如，以x = 0为中心的sin(x)的Taylor's series为：sin(x) = x - (x^3)/3! + (x^5)/5! - (x^7)/7! + ...类似地，cos(x)的Taylor's series为：cos(x) = 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + ...这些级数使我们能够以极高的精度计算三角函数，尤其是在仅使用级数的少数项时。在实际应用中，Taylor's series被广泛应用于物理、工程和计算机科学等领域，在这些领域中，近似函数对于解决现实问题至关重要。然而，需要注意的是，Taylor's series有其局限性。级数的收敛性取决于函数及其展开的点。对于某些函数，Taylor's series可能仅在某个区间内收敛，而对于其他函数，则可能完全发散。因此，在应用Taylor's series时，理解收敛半径至关重要。总之，Taylor's series是数学中不可或缺的工具，促进了复杂函数的近似。它将函数表示为无限多项式项之和的能力为分析和计算开辟了无数可能性。随着我们继续探索数学分析的深度，Taylor's series仍然是一个基本概念，连接着各个研究领域，使我们能够更轻松、更准确地解决复杂问题。