# FSU-MATH2400-Project6

Author

Sarah Wright

Last Updated

5 years ago

License

Creative Commons CC BY 4.0

Abstract

In this calculus project, students use infinite series to investigate Euler's Equation: $e^{i\pi} + 1 = 0$.

Author

Sarah Wright

Last Updated

5 years ago

License

Creative Commons CC BY 4.0

Abstract

In this calculus project, students use infinite series to investigate Euler's Equation: $e^{i\pi} + 1 = 0$.

```
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\setlength{\parindent}{15pt}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage[colorlinks = true, linkcolor = black, citecolor = black, final]{hyperref}
\usepackage{graphicx}
\usepackage{multicol}
\usepackage{ marvosym }
\usepackage{wasysym}
\usepackage{tikz}
\usetikzlibrary{patterns}
\newcommand{\ds}{\displaystyle}
\DeclareMathOperator{\sech}{sech}
\setlength{\parindent}{0in}
\pagestyle{empty}
\begin{document}
\thispagestyle{empty}
{\scshape Math 2400} \hfill {\scshape \large Euler's Equation} \hfill {\scshape Project \#6}
\smallskip
\hrule
\bigskip
Often referred to as the most beautiful equation in mathematics, Euler's Identity, $$e^{i\pi} + 1 = 0$$ involves the five most important constants; the additive identity 0, the multiplicative identity 1, the imaginary number $i$, and the two irrational numbers $e$ and $\pi$. It should feel crazy that this is true! The goal of this project is to see {\bf why} this equation works and use infinite series in a new way.
\bigskip
You should present your work on this project in a written format for a reader learning about infinite series for the first time; think about yourself at the start of this chapter of material. Explain what infinite series are, convergence vs. divergence, and how series can represent functions.
\bigskip
You want to show that $e^{i\pi} + 1 = 0$. The main idea is to use infinite power series to show Euler's Formula $$e^{i\theta} = \cos\theta + i\sin \theta.$$
\bigskip
The imaginary number $i$ that appears here may be new to you. The only fact you really need for this is that $i^2 = -1$. We can multiply $i$ by real numbers as well as $i$ itself where simplifications can be made. For example:$$(-2i)^3 = (-2)^3i^3 = -8(i^2)(i) = -8(-1)i = 8i$$
\bigskip
You do not need to delve into the world of complex analysis\dots So, to check convergence in this assignment, it is safe to assume that the absolute value of $i$ is one, and adjustments can be made accordingly if needed. So, $$\left|-\frac{1}{2}i\right| = \left|-\frac{1}{2}\right|\left|i\right| = \frac{1}{2}$$
\bigskip
It is up to you to organize this work and your explanation in a logical way. Keep the intended audience in mind as well as the guidelines for written work found in the {\it Specifications for Calculus Work} document.
\end{document}
```

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