The first calculus section on sequences is here. Sequences aren't entirely the most interesting subject in calculus because they're little more than functions that only take integers. The more interesting kind of sequences are sequences whose terms references previous terms. These are called recurrence relations and are studied in more detail in differential equations. Anywho, for now, vanilla calculus sequences are now available.
What's the best subject to start learning proof-based mathematics? With a dizzying array of subjects to pick from, it's hard to tell which you're ready for and which are specialized subfields of specialized subfields.
Most college curricula offer real analysis, complex analysis, number theory, abstract algebra, and probability theory as introductory proof-based classes. Others may offer a proof-based approach to linear algebra, and more rigorous curricula may even include things like functional analysis. While all of these subjects form the bedrock of higher mathematics, it is my opinion that none of them is well-suited to teaching proofs per se.
In a previous post, I explained why proofs are so difficult to understand. In this post, I'd like to explain how to think about them productively.
A proof takes given axioms, theorems, and other mathematical properties and combines them in a way that demonstrates the truth of a new mathematical property. The first step in conquering proofs is to acquire a new mindset about solving problems, one that is better suited to writing proofs rather than solving equations. The next step is to understand exactly which logical moves are permitted by the given knowledge, and just as importantly those moves that are not. The final step is to acquire a toolbox of proof techniques that enable you to more easily solve an ever-growing suite of problems.
Oh boy! Integrals IV is now live! This one's got all the juicy integration techniques in it. All of the previous sections have been leading up to this one. This is the meat and potatoes, folks. We've got your u-substitution and your integration by parts, and we've even got your trigonometric substitution and your partial fractions. These are the methods generations of mathematicians have learned and applied time and again for both business and pleasure, time and time again.
I hope that these sections in particular can become great repositories of interesting problems over time. Many problem sets, textbooks, and Q&A sites contain all kinds of interesting examples, and I'd like to include as many of them as I can all in one place. For now, each section has at least 10 problems, but someday I'd like them to be definitive collections.