In this calculus project, students use infinite series to investigate Euler's Equation: $e^{i\pi} + 1 = 0$.
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{\scshape Math 2400} \hfill {\scshape \large Euler's Equation} \hfill {\scshape Project \#6}
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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.
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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.
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You want to show that $e^{i\pi} + 1 = 0$. The main idea is to use infinite power series to show Euler's Formula $$e^{i\theta} = \cos\theta + i\sin \theta.$$
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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:$$(-2i)^3 = (-2)^3i^3 = -8(i^2)(i) = -8(-1)i = 8i$$
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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, $$\left|-\frac{1}{2}i\right| = \left|-\frac{1}{2}\right|\left|i\right| = \frac{1}{2}$$
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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.
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