“i” Love π Flavored Series By David Monroe and Daniel Newton

BEFORE WE BEGIN Don’t worry, we will not go crazy in depth Feel free to do research on your own afterwards Save “bigger picture” questions until our quick break

Introduction: π, i, and What are they? What are their values? Does the square root of a negative number have use? Why mention them together? Why does e^(i*π)+1=0? Pi is the constant found when the circumference of a circle is divided by its diameter. It is estimated to be 3.14159265. i is the constant used when taking the square root of negative 1. Square roots of negative numbers aren’t real. We mention them together in order to find the equation listed above.

What are series, and why are they important? Are infinite series infinite, and what does it mean to converge? Series are a summation of functions and are important because they are easier to use than other functions Convergence means the terms are getting small enough that we do not get infinity Ex of divergence and convergence: Gabriel’s horn/Geometric series

How do they relate to series? Derivatives So what is a derivative? How do they relate to series? A derivative is the instantaneous slope of a function at a particular point on the graph. With Taylor series, we take the nth derivative of the function for the polynomial expansion. Explain why e^x’s derivative is e^x.

Why do we care about them? Does this relate to derivatives or series? Trigonometry Another thing to learn? Why do we care about them? Does this relate to derivatives or series? Trigonometry is the study of angles. We use sin, cos, and tan to explain how the angles of shapes relate to one another. We care about them because sin, cos, and tan become very important in derivatives and series.

Questions and Cookie Pizza π What questions do you have so far? Let us discuss them over a slice of π! 10-15 minutes

Piecing it all together Relating this to e, π, and i Back to Series Piecing it all together Relating this to e, π, and i https://www.desmos.com/calculator/g okjepwgi6 We can approximate functions with derivatives. Explain factorials quickly. And obviously, piece all of our topics together. The link goes to show the taylor series of sine cosine and e, here we show how these polynomials can add up to our functions.

Taylor Series What make a series a Taylor series? How does this relate sin and cosine to this?

Common Taylor Series Here are common taylor series

The Derivation (The Main Event!) Here is our culmination of everything we have learned. This is where we prove e^(i*pi)+1=0.

So what? What else? Final thoughts? Discussion Series are easier to work with compared to complicated functions or functions we don’t know For e^i(theta) we use it for rotation in 2D, Fun fact if we want rotation in 3D with a similar method we need four dimensions