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Let’s Get Real… Analysis (Part 10): Cool Stuff with Cauchy Sequences
What does it mean for a sequence to be a Cauchy sequence? Why are Cauchy sequences important? Great questions! Hopefully by the end of this article you can answer these for yourself! The Need for More Recall back in Let’s Get Real… Analysis…
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Let’s Get Real… Analysis (Part 9): limsup’s, liminf’s, and the Bolzano-Weierstrass Theorem
We’ve asked and answered the question, “Can we ever prove that a sequence must converge without needing to know or find its limit beforehand?” in Let’s Get Real… Analysis (Part 5): The Monotone Convergence Theorem. Today, we will be proving something that has…
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Let’s Get Real… Analysis (Part 8): Superb Subsequence
Sometimes we have sequences that don’t converge, and yet, it’s possible to take elements from the divergent sequence to form a convergent sequence. We refer to such sequences as subsequences, which will be the topic of interest today. But you knew that, you…
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Let’s Get Real… Analysis (Part 7): Limit Laws for Sequences
We’re continuing in our Let’s Get Real… Analysis series mission to gather more tools for analyzing sequences. Today, we will learn what I call, our limit laws. These will likely be intuitive, but they give us an opportunity to practice writing proofs, which…
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Let’s Get Real… Analysis (Part 6): The Ratio Test for Sequences
Once you learn how to use the definition of a sequence limit to prove that a sequence converges to a limit, the next step is to develop a set of tools that you can use so that way you can solve more challenging…
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What Happens When You Add Infinitely Many Numbers Together???
Question: What Happens When You Add Infinitely Many Numbers Together??? I’ve heard that when you ask people this question, their answer tends to be something like, “Well, you get infinity!” However, this is not always the case! For instance, you can add infinitely…
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Trig Rules the Old-Fashioned Way
Last week we learned some tricks to derive some trigonometry identities quickly while using Euler’s formula; however, once I finished writing it, I wanted to prove the same rules using triangles and geometry! I thought it would be fun and boy howdy was…
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Euler’s Formula as a Tool for Fast Trigonometry
Preamble (Feel free to skip) There are many times (at least in school) where we need some random trigonometric identity to be able to solve a problem. At least for me, I’ve never bothered to memorize the double angle formula or the sum…
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How to Write a Proof by Contradiction
There are many different methods one can use when trying to prove a mathematical statement, such as induction, using the contrapositive, or other direct proof methods. Today we discuss is known as a proof by contradiction and is an incredible tool to have…
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Protected: A Kick in the Mystery
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What Comes Next: 1, 2, 4, 8, 16, … hint it’s not 32
Hello,
Welcome to A Kick in the Discovery! If you’re curious where this phrase came from, that’s good! It came from the great physicist Richard Feynman. He was curious about anything and everything and he loved to explain science to others. I want to communicate the same enthusiasm about things that I find amazing! More than that, I want to practice teaching and explaining more and more complex ideas. Communication is very important and very challenging, and I hope this blog helps me learn this skill. With all this said, please bear with me while I practice my writing skills! It will be a process.
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