关于调和级数的部分和的推导及证明

Derivation and Proof of Partial sum of Harmonic Series

中世纪后期,数学家Oresme证明了所有调和级数都是发散的,但是调和级数的拉马努金和存在,且为Euler常数.Euler在1734年利用Newton的成果,首先给出了调和级数的部分和的表达式.通过分析Ross,S.M.对经典概率论问题"优惠券收集问题"的解决方法,得到了调和级数的部分和的不同表达式,并运用数学归纳法,变量代换证明了表达式的正确性.

In the late Middle Ages,the mathematician oresmee proved that all harmonic seri-es are divergent,but the ramanugin sum of harmonic series exists and is Euler constant.In 1734,Euler first gave the expression of the partial sum of harmonic series by using Newton’s results.In this paper,by analyzing Ross,S.M.’s solution to the classical probability problem"coupon collection problem",different expressions of the partial sum of harmonic series are obtained,and the correctness of the expressions is proved by mathematical induction and variable substitution.