Calculus I

Patterns That Go Somewhere

Or: What does a list of numbers want to become?

Companion notebook. Play with this chapter’s ideas — sliders, animations, and drills — in notebooks/ch1_sequences.ipynb.

By the end of this chapter you will be able to look at a never-ending list of numbers and answer a question that turns out to be the seed of all of calculus: what number, if any, is this list sneaking up on?

You will recognize the two great families of sequences — the ones that grow by adding the same amount each step, and the ones that grow by multiplying by the same amount — and you will have formulas for both. That part is mostly a sharpening of patterns you already met in Book 1.

Then comes the genuinely new idea, the one this whole book is built on. Some lists settle down. The list $1, \tfrac12, \tfrac13, \tfrac14, \ldots$ is creeping toward $0$ without ever arriving. The list $2,\ 2.25,\ 2.37,\ldots$ is creeping toward a strange number near $2.718$. Pinning down what “creeping toward” means — precisely enough that we can prove it rather than just squint at it — is the idea of a limit, and once you have it, derivatives and integrals are within reach.

And you will learn the part that separates people who know about limits from people who can actually use them: what to do when the obvious move fails. By the end you’ll be able to evaluate limits where naive substitution gives nonsense like “$\infty$ times $0$” — where the answer is decided by a hidden race between two parts of the expression — and you’ll meet a stubborn $\infty\cdot 0$, the $n$-th root of $n$, and the whole family of $e$, each of which looks impossible until you know the trick.

A note on the two layers (please read this once)

Starting in this book, every chapter runs on two layers, and you should know which one you are walking.

Layer 1 is the main road. It is the ideas, the pictures, and how to actually compute things. Everyone walks the main road, start to finish. If you read only Layer 1 you will understand what is going on and be able to do the work.

Layer 2 is the set of marked detours — the boxes labeled with a diamond, ◆. This is where we stop pointing at a picture and prove the thing is really true, using exact definitions. On your first pass through a chapter you are welcome to glance at these and keep walking. On your second pass — or right now, if you are hungry — the climbs are the best part, and they are where you stop taking my word for anything.

Why the proofs start here, in Chapter 1, when Book 1 mostly let you trust your eyes? Because a limit is the first idea in this series that is genuinely slippery — your intuition about “getting close” is good enough to get started and not good enough to keep you safe later. So this is exactly where proof begins to earn its place. Let’s go.

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