Sequences

A sequence is simply a list of numbers that goes on forever, like $1, \frac12, \frac13, \dots$, made by some rule. It's good for describing anything that happens in steps: a bank balance year by year, a drug level dose by dose, or each new guess an algorithm makes. All we're really asking about it is what number the list is closing in on. The formula $\lim_{n\to\infty} a_n = L$ looks formal, but it's just three small pieces: $a_n$ (the number sitting in position $n$ of the list), $n \to \infty$ (go farther and farther down the list) and $L$ (the number the list ends up hugging). Put together it says "far enough down the list, the entries are basically $L$," so bear with the notation.

A sequence is just a list that never ends. You can give it by a formula, like $a_n = \dfrac{n}{n+1}$, which lists $\frac12, \frac23, \frac34, \dots$, or recursively, where each term is built from the one before, like $a_1 = 1$, $a_{n+1} = 2a_n + 1$. The whole sequence is written $\{a_n\}_{n=1}^{\infty}$.

Convergence. Look at $\frac12, \frac23, \frac34, \dots$ for a moment. The terms keep creeping toward 1, and once you're far enough out they're as close to 1 as you want and they stay there. That's exactly what it means for a sequence to converge to $L$: the terms get, and stay, as close to $L$ as you like once $n$ is large enough. If no such $L$ exists, the sequence diverges, either by growing without bound or by never settling down, like $1, -1, 1, -1, \dots$

Computing limits. Here's the good news: you already know how to do most of these. A sequence $a_n = f(n)$ is just the function $f$ sampled at $1, 2, 3, \dots$, so if $f(x) \to L$ as $x \to \infty$, the samples have no choice but to head to $L$ too. That means every Calculus I tool carries over, including L'Hôpital's rule. Just apply it to $f$, since you can't differentiate a sequence: it only lives at whole numbers.

Facts you'll use constantly:

• $r^n \to 0$ if $|r| < 1$, and $r^n \to 1$ if $r = 1$. For every other $r$ the sequence $r^n$ diverges.
• If $|a_n| \to 0$ then $a_n \to 0$. This handles signs like $(-1)^n$.
• Squeeze theorem: if $b_n \le a_n \le c_n$ and both $b_n$ and $c_n$ approach $L$, so does $a_n$.
• Continuous functions pass through limits: if $a_n \to L$, then $f(a_n) \to f(L)$.

A common mistake. It's tempting to think that if $f(x)$ has no limit, then the sequence $f(n)$ can't have one either. It can. $\sin(\pi n) = 0$ for every whole number $n$, so that sequence is $0, 0, 0, \dots$ and converges to 0, even though $\sin(\pi x)$ keeps waving forever as $x \to \infty$. The sequence only sees the whole numbers, and those happen to land on the zeros. So the function trick works in one direction only: a function limit gives you the sequence limit, but a missing function limit tells you nothing.

Iterative algorithms produce sequences. Newton's method generates $x_1, x_2, \dots$ that should converge to a root, and a training loop produces a sequence of losses. Asking whether they converge, and how fast, is asking about sequence limits.