带皮亚诺余项和拉格朗日余项的泰勒公式应用比较

Comparison of Taylor Formula with Peano and Lagrange Remainders

本文给出了带拉格朗日余项和皮亚诺余项的泰勒公式在应用上的比较,带皮亚诺余项的泰勒公式可用于求极限、高阶导数、无穷小阶的判定等,而带拉格朗日余项的泰勒公式可用于证明适合某种条件的存在性、不等式的证明、方程根的问题、近似计算等.

Taylor formula with Lagrange remainder and Taylor formula with Peano remainder are compared through the applications. Peano remainder works well with limits, higher order derivatives, and order of infinitesimals, etc., while Lagrange remainder can be used to prove the existence of certain conditions, inequalities, solutions of equations, and approximations, etc.