应用黎曼-斯蒂尔杰斯积分证明黎曼第二积分中值定理（英文）

黎曼第二积分中值定理是数学分析中的重要结论,在反常积分和级数理论中都有重要的应用,但它同时又是数学分析的教学难点。华东师范大学《数学分析》教材中黎曼第二积分中值定理的证明略显复杂,极大地掩盖了证明的本质。应用黎曼-斯蒂尔杰斯积分其中的分部积分公式重新证明了黎曼第二积分中值定理,该证明方法简单易懂,还可以应用到其他定理(如反常积分中的狄里克雷判别法等)的证明。

Proving the Second Mean Value Theorem for Riemann Integrals by Riemann-Stieltjes Integrals

The second mean value theorem for Riemann integrals is an important result in mathematical analysis with many applications in the theories of improper integrals and series.On the other hand,it is also a difficulty for teaching.In the text book 《 Mathematical Analysis》 published by East China Normal University,the proof of the theorem is quite complicated from which the essential of the proof is covered to a large extent.Proving the second mean value theorem for Riemann integrals by integrating by parts formula of Riemann-Stieltjes integrals is much more simpler,and is easy to be used to prove other results such as the Dirichlet criteria for improper integrals.