Deleting Items and Disturbing Mesh Theorems for Riemann Definite Integral and Their Applications
Provides theoretical extensions to Riemann integration for mathematicians working on advanced limit problems.
The paper presents theorems on deleting items and disturbing the mesh of partitions in Riemann sums, showing that under specific conditions the limit still converges to the Riemann definite integral. These results help solve complex limits of sums and products of series.
Based on the definition of Riemann definite integral,deleting items and disturbing mesh theorems on Riemann sums are given. After deleting some items or disturbing the mesh of partition, the limit of Riemann sums still converges to Riemann definite integral under specific conditions. These theorems can deal with a class of complicate limitation of sum and product of series of a function, and demonstrate that the geometric intuition of Riemann definite integral is more profound than ordinary thinking of area of curved trapezoid.