A small Wolfram Mathematica project working on proving the Taylor series and Euler's Identity.
Problem Statement:
For this project I tried to look at the ways that mathematicians use series to approximate polynomial functions such as trigonometric functions. I became interested in this topic after hearing about how physicists used Euler’s identity to represent waves that would usually be represented by trigonometric equations. I looked into how this worked and found something called a Taylor series. I specifically explored the Taylor series as a polynomial approximation of function. I tried to prove the Taylor series using computing through Wolfram Mathematica. The goal of this exploration was to prove that given some function g(x), if i(x) was the Taylor series of that function, as the size of the series reaches infinity, i(x) is equal to g(x). To accomplish this I learned how to use Wolfram Mathematica and the basics and applications of the Taylor series.
Process and Solution: Taylor Series Proof
Applications and Extensions:
For my extension I pursued the proof of Euler’s identity using Wolfram Mathematica. The applications for the Taylor series are mainly in quantum mechanics. The Taylor series can be used to prove Euler’s identity. Euler’s identity states that eix=cos(x)+i sin(x). This is because the sum Taylor expansions of cosine and sine matches the pattern of the Taylor series of eix. Euler’s identity has applications in the representation of light in quantum mechanics. When Euler’s identity is graphed in the complex plane it forms a function that can be manipulated to represent waves of light.