Part 1/5:
Understanding the McLaren Series and Convergence Intervals
In mathematical analysis, the McLaren series, also known as the Taylor series centered at zero, provides powerful tools for approximating functions through polynomial expressions. This article explores the McLaren series for a selection of functions, detailing their respective series representations and intervals of convergence.
Part A: The Function ( \frac{1}{1 - x} )
The McLaren series for the function ( f(x) = \frac{1}{1 - x} ) is derived from the geometric series. This series can be expressed as:
[
f(x) = \sum_{n=0}^{\infty} x^n
]
The interval of convergence for this series is found where ( |x| < 1 ). Therefore, the radius of convergence is ( R = 1 ).