The Proof of Taylor Formula and Its Application

Taylor formula is the embodiment of approximatioss of calculous and has important applications in various aspects of calculus.This paper introduces a simpler new method of proof of Taylor formula on the basis of theorems,which have been stated in current general textbooks,and its proofs.In view of extensive use of Taylor formula in analysis and study of problems in mathematical analysis and solutions of practical applications,the paper have generalized applications in nine aspects which include limit and differential coefficient calculation,judgement of convergence and divergence of progression and improper integral,proof of inequality,proof of definite integral,determinant calculation and mean-value formula,mean-value estimate of derivative,boundary estimate,and illustrated relevant techniques of applications.