级数的重排以及收敛性

Rearrangement of Series and Convergence

摘要: 无穷级数理论是微积分学的重要内容，它在数学的许多领域都有着广泛的应用，例如函数表示等。本文通过无穷级数的重排问题，论述了绝对收敛级数的重排和条件收敛级数重排的相关理论，同时，阐述了交错级数${\displaystyle {\sum }_{n=1}^{\infty }{\left(-1\right)}^{n-1}}/n$ 的重排，给出了交错级数${\displaystyle {\sum }_{n=1}^{\infty }{\left(-1\right)}^{n-1}}/\sqrt{n}$ 的收敛条件。

Abstract: The theory of infinite series is an important content of calculus, which has a wide range of applications in many fields of mathematics, such as function representation and so on. In this paper, through the problem of rearrangement of infinite series, the related theories of rearrangement of absolutely convergent series and rearrangement of conditionally convergent series are discussed, meanwhile, the rearrangement of ${\displaystyle {\sum }_{n=1}^{\infty }{\left(-1\right)}^{n-1}}/n$ intersecting series is elaborated, and the convergence condition of ${\displaystyle {\sum }_{n=1}^{\infty }{\left(-1\right)}^{n-1}}/\sqrt{n}$ intersecting series is given.