10.2 Sequences
Sequences lie at the very heart of mathematics. Sequences and their limits describe long-term behavior, like the (occasional) convergence of gradient descent to a local optimum. By definition, a sequence is an enumeration of mathematical objects.
The elements of a sequence can be any mathematical object, like sets, functions, or Hilbert spaces. (Whatever those might be.) For us, sequences are composed of numbers. We formally denote them as
For simplicity, the subscripts and the superscripts are often omitted, so don’t panic if you see {an}, as it is just an abbreviation. (Or an. Mathematicians love abbreviations.) If all elements of the sequence belong to a set A, we often write {an}⊆A.
Sequences can be bidirectional as well. Those are denoted as {an}n=−∞∞. We don’t need them for now, but they will frequently come up in the context of probability distributions later.
Sometimes we don’t need an entire sequence, just a...
