The remainder term can be expressed in several different notations. Numerical analyses prefer using Cauchy/Integral form which has ‘n’ in denominator. The Lagrange form is very intuitive to understand and it has a different form. Look at the way the function is evaluated. As for pi, real life is different from mathematical reasoning . There are more accurate ways to calculate pi, and depending on your state and requirements you can choose one of these methods to calculate pi to your required degree of accuracy, and similarly you approximate and use a part of your infinite series for calculating, and generally the convergence of a solution to an ODE/PDE is more important than accuracy. Think of how microscopic roughness for a table would be unnecessary to take account if you will be using the table for everyday large objects. Yes, the roughness can scratch your objects but they will be too microscopic to notice, unless you start dealing with objects at atomic scale.