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高等数学中的幂级数知识是数学学科中的重要部分，广泛应用于物理、工程、经济学等领域。幂级数在实际应用中有很多的优势，如能够表示函数的解析式、求出函数的导数和积分等。今天，我们将为大家介绍一下幂级数的基本概念和运算法则。

The knowledge of power series in advanced mathematics is an important part of the mathematical discipline and is widely used in physics, engineering, economics and other fields. Power series have many advantages in practical applications, such as being able to represent the analytical formula of functions, finding derivatives of functions, and integrals. Today, we will introduce you to the basic concepts and algorithms of power series.

什么是幂级数？

What is a power series?

幂级数是无穷级数的一种，表现形式为：f(x) = a0 a1(x-a) a2(x-a)² ..., 其中a0, a1, a2, ...是常数，称为幂级数的系数，x是变量，a是常数，称为幂级数的中心。幂级数在中心周围有一个收敛半径，幂级数在其收敛半径内绝对收敛，否则绝对发散。

A power series is a type of infinite series expressed as: f(x) = a0 a1(x-a) a2(x-a)² ..., where a0, a1, a2, ... is a constant, called the coefficient of the power series, x is the variable, and a is the constant, called the center of the power series. The power series has a radius of convergence around the center, and the power series absolutely converges within its radius of convergence, otherwise it absolutely diverges.

常用运算法则

Common algorithms

1. 幂函数的和的幂级数：设f(x)=∑n=0∞an(x-a)和g(x)=∑n=0∞bn(x-a)，则它们的和f(x) g(x)=∑n=0∞(an bn)(x-a)。

1. The power series of the sum of power functions:

let f(x)=∑n=0∞an(x-a) and g(x)=∑n=0∞bn(x-a),

then their sum f(x) g(x)=∑n=0∞(an bn)(x-a).

2. 幂函数的积的幂级数：设f(x)=∑n=0∞an(x-a)和g(x)=∑n=0∞bn(x-a)，则它们的积f(x)·g(x)=∑n=0∞∑k=0nan-kbk(x-a)。

2. Power series of products of power functions:

let f(x)=∑n=0∞an(x-a) and g(x)=∑n=0∞bn(x-a),

then their product f(x)·g(x)=∑n=0∞∑k=0nan-kbk(x-a).

3. 幂函数的复合的幂级数：设f(x)=∑n=0∞an(x-a)和g(x)=∑n=0∞bn(x-a)，则它们的复合f(g(x))=∑n=0∞an{b0 b1(x-a) b2(x-a)² ... bn(x-a)^n ...}。

3. The power series of the composite of the power function:

let f(x)=∑n=0∞an(x-a) and g(x)=∑n=0∞bn(x-a),

then their composite f(g(x))=∑n=0∞an{b0 b1(x-a) b2(x-a)² ...

bn(x-a)^n ...}.

幂级数在复杂的数学问题中发挥着重要作用，如在微积分、常微分方程、偏微分方程和物理学中应用广泛。了解幂级数知识，可以帮助我们更好地理解数学模型，在实际问题中更好地运用数学方法解决问题。

Power series play an important role in complex mathematical problems, such as calculus, ordinary differential equations, partial differential equations, and physics. Understanding the knowledge of power series can help us better understand mathematical models and better use mathematical methods to solve problems in practical problems.

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参考资料：百度百科-秒懂百科、《高等数学辅导及习题精解》、《高等数学第七版上册》

翻译：有道翻译

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文字|歪歪

排版|歪歪

审核|杨德鸿

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