Chapter 6 Sequences & series of real numbers

(last updated: 4:08:23 PM, February 18, 2026)

6.1 Sequences

A sequence of a function whose domain is the natural numbers \(f(n)\) with \(f:\mathbb N\to\mathbb R\). An example is \(f(n)=\frac1n\) which gives the sequence \(1,\frac12,\frac13,\ldots\).

Often we use subscript notation instead: \(x_n=f(n)\) or \(a_k=f(k)\). It is typical to use some kind of bracket notation too: \((x_n)_{n\in\mathbb N}\) or \(\{x_n\}_{n\in\mathbb N}\). I prefer using parentheses and not curly braces as I like to restrict curly braces for sets. A sequence is not a set. Here is another common notation: \((x_n)_{n=1,2,\ldots}\).

We’ll discuss convergence of sequences first… (under construction…)

6.1.1 Subsequences

6.1.2 Monotone sequences

6.1.3 Limsup and liminf

6.2 Series

A finite series is \(\sum_{j=1}^n a_j=a_1+a_2+\cdots+a_n\) and and infinite series is \(\sum_{j=1}^n a_j=a_1+a_2+\cdots+a_n+a_{n+1}+\cdots\) and keeps going on forever.