Question

What is the Taylor Series expansion? Why do they exist and why are they important?

Answer 1

Taylor series are used to represent functions as an infinite sum (of polynomials) using the function's derivatives. SOmetimes they are useful because we can use a finite sum of the series to approximate a very complicated function that may be hard to integrate... but the polynomial terms of the Taylor series are easy to integrate. It can sometimes also help determine the asymptotic behaviour of a function. Also, computer software may use Taylor series to calculate approximate values for certain numerical values (like finding the value of \(e \) to 20 decimal places for example) or to find the value of equations that can't be determined analytically.