Taylor Formula and Its Application Problem

Taylor formula and Taylor series are particularly important in mathematical analysis teaching and are tools for numerical calculation. Their applications are very extensive, including calculating limits, indefinite integrals, definite integrals, high - order derivative values, determining the series are convergent or divergent, proving inequalities, approximate treatment, solving differential equations, and proving identities. This paper mainly introduces the definition of Taylor’s formula, the formula along with related proof, and the application of approximate calculation, indefinite integral, inequality proof. It was developed in the study of function expansions to provide a fine description of how a function behaves near a certain point. The significance of Taylor’s formula is that it provides a way to expand a function into an infinite series, so that the properties of a function near a certain point can be studied. It can increase the quality of teaching as well as broaden the scholastic problem - solving ideas and improving their ability to solve problems.