Abstract
This chapter discusses one of the most important applications of differentiation, which concerns infinite series. It focuses on the convergence of infinite series. The chapter discusses Taylor's theorem, which provides an estimate of the error made in a polynomial approximation to a function. Maclaurin's theorem is a special case of Taylor's theorem, which gives an expansion of a function “about the origin. Taylor's theorem is a generalization of the mean value theorem. Important functions of mathematics can be expressed as infinite power series by the means of Maclaurin's or Taylor's theorems. The determination of maxima and minima is important in physical problems. The most economical use of material involves an investigation of the minima of some function. The nature and stability of the equilibrium states of mechanical and physical systems are determined by finding the conditions under which some functions, such as potential energy, strain energy, and entropy, have a maximum or a minimum.
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