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无穷级数
作为高数部分的考试重难点,根据考研大纲要求,它的考试内容包括:
01
考试内容
常数项级数的收敛与发散的概念收敛级数的和的概念级数的基本性质与收敛的必要条件几何级数与级数及其收敛性正项级数收敛性的判别法交错级数与莱布尼茨定理任意项级数的绝对收敛与条件收敛函数项级数的收敛域与和函数的概念幂级数及其收敛半径、收敛区间(指开区间)和收敛域幂级数的和函数幂级数在其收敛区间内的基本性质简单幂级数的和函数的求法初等函数的幂级数展开式函数的傅里叶(Fourier)系数与傅里叶级数狄利克雷(Dirichlet)定理函数在上的傅里叶级数函数在上的正弦级数和余弦级数。
02
考试要求
1.理解常数项级数收敛、发散以及收敛级数的和的概念,掌握级数的基本性质及收敛的必要条件 2.掌握几何级数与级数的收敛与发散的条件
3.掌握正项级数收敛性的比较判别法和比值判别法,会用根值判别法 4.掌握交错级数的莱布尼茨判别法
5.了解任意项级数绝对收敛与条件收敛的概念以及绝对收敛与收敛的关系 6.了解函数项级数的收敛域及和函数的概念
7.理解幂级数收敛半径的概念,并掌握幂级数的收敛半径、收敛区间及收敛域的求法
8.了解幂级数在其收敛区间内的基本性质(和函数的连续性、逐项求导和逐项积分),会求一些幂级数在收敛区间内的和函数,并会由此求出某些数项级数的和
9.了解函数展开为泰勒级数的充分必要条件 10.掌握麦克劳林(Maclaurin)展开式,会用它们将一些简单函数间接展开为幂级数
11.了解傅里叶级数的概念和狄利克雷收敛定理,会将定义在上的函数展开为傅里叶级数,会将定义在上的函数展开为正弦级数与余弦级数,会写出傅里叶级数的和函数的表达式。
补充公式与结论
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