Sequences and Series

A sequence is an arrangement of numbers in a definite order, formed according to some definite rule. Formally, a sequence is a function $f$ whose domain is the set of natural numbers $\mathbb{N}$ (or, for a finite sequence, the first $n$ natural numbers); the value $f(n)$ is written $a_n$ and called the $n$th term (or general term) of the sequence. The sequence itself is written $a_1, a_2, a_3, \ldots, a_n, \ldots$, or briefly $\{a_n\}$.

Finite and infinite sequences. A sequence with a last term is a finite sequence; one that goes on forever, with a term for every natural number, is an infinite sequence. Unless stated otherwise, "sequence" in this chapter means an infinite sequence, though every result applies equally to the first $n$ terms of one.

Describing a sequence. A sequence can be specified in two ways:

• Explicitly, by a formula for $a_n$ directly in terms of $n$. For example, $a_n = 2n-1$ generates the sequence of odd numbers $1, 3, 5, 7, \ldots$, since $a_1=1, a_2=3, a_3=5,$ and so on.
• Recursively, by giving one or more starting terms together with a rule that produces each later term from the ones before it. For example, the Fibonacci sequence is defined by $a_1=1, a_2=1$, and $a_n = a_{n-1}+a_{n-2}$ for $n \ge 3$, giving $1,1,2,3,5,8,13,\ldots$

Series. If $a_1, a_2, a_3, \ldots, a_n, \ldots$ is a sequence, the expression $a_1+a_2+a_3+\cdots+a_n+\cdots$ formed by adding its terms is called the series associated with that sequence. The sum of the first $n$ terms of the series is denoted $S_n$ and is written compactly using sigma (summation) notation:

A sequence is an ordered list; a series is the sum of that list — the two words are often used loosely for each other in everyday speech, but in mathematics they name different objects and should not be confused.

The relation between $a_n$ and $S_n$. Since $S_n = a_1+a_2+\cdots+a_{n-1}+a_n = S_{n-1}+a_n$, it follows that for every $n \ge 2$,

This identity is used constantly: whenever the sum $S_n$ of a series is known as a formula in $n$, the individual term $a_n$ can always be recovered by subtracting consecutive sums, without needing any other information about the sequence.

Progressions. A sequence whose terms follow a specific, predictable mathematical pattern — so that any term can be located by a formula rather than merely by continuing the list — is called a progression. This chapter studies the two most fundamental progressions, the Arithmetic Progression (A.P.), where consecutive terms differ by a constant amount, and the Geometric Progression (G.P.), where consecutive terms are in a constant ratio; a natural combination of the two, the Arithmetic-Geometric Progression (A.G.P.), whose terms multiply an A.P. by a G.P. term-by-term; the special case of an infinite G.P. whose terms shrink toward zero, which can be meaningfully summed even though it never ends; and, finally, three special sums — of the first $n$ natural numbers, their squares, and their cubes — that recur constantly, both within this chapter and throughout the rest of the syllabus.