V2EX harmonic series

Harmonic Series

释义 Definition

harmonic series调和级数。指数学中一个著名的无穷级数:
\[ 1+\frac12+\frac13+\frac14+\cdots \] 它的部分和增长得很慢,但最终会发散(不收敛)。在物理/音乐语境中,“harmonic series”也常指泛音列(与频率成整数倍关系的一系列谐波);此处以数学义为主。

发音 Pronunciation

/hrmnk sriz/
/hmnk sriz/

例句 Examples

The harmonic series diverges.
调和级数是发散的。

Although the harmonic series grows very slowly, its partial sums can exceed any number if you add enough terms.
尽管调和级数增长得很慢,但只要加上足够多项,它的部分和就能超过任何给定的数。

词源 Etymology

harmonic 来自希腊语 harmonia(“和谐、协调”),经由拉丁语和法语进入英语;在数学里引申为“与比例/整数关系相关、看起来‘协调’的”。series 来自拉丁语 series(“一串、连续排列”)。合在一起,“harmonic series”字面即“调和的串列/级数”,用于指这种与调和平均、对数增长等性质密切相关的经典级数。

相关词 Related Words/h2>

文学与名著中的用例 Literary Works

  • G. H. Hardy, Divergent Series(讨论调和级数作为典型发散级数的例子)
  • Tom M. Apostol, Calculus, Volume 1(用调和级数说明级数的收敛与发散)
  • Konrad Knopp, Theory and Application of Infinite Series(在无穷级数理论中频繁出现)
  • Leonhard Euler, Introductio in analysin infinitorum(早期分析学经典著作中涉及相关级数与对数增长思想)
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