Three Important Taylor Series for Introductory Physics

摘要

Taylor expansions of the exponential exp(x), natural logarithm ln(1+x), and binomial series (1+x)n are derived to low order without using calculus. It is particularly simple to develop and graph the expansions to linear power in x. An example is presented of the application of the first- order binomial expansion to finding the electrostatic potential at large distances from an electric dipole. With a little extra work, the second-order expansions can be obtained starting from the familiar kinematics expression for the motion of a particle accelerating in one dimension, which instructively ties the mathematical development to physics concepts already presented in introductory courses.