Sequences

Overview

A function $f$ whose domain is the set of natural numbers is called an (infinite) sequence. The function value $f (n)$ is called the $n$ th term of the sequence. Sequences are described using one of:

Closed Formulas. A formula that can be computed using a fixed number of "well-known" standard operations.
Recursive Definitions. Consists of initial conditions and a recurrence relation.

We often denote a sequence as $(a_{n})_{n \geq 0}$ where $a_{i} = f (i)$ for all $i \in N$ . Other index patterns are denoted in the natural way.

A tail of $(a_{n})$ refers to the sequence with a finite number of initial terms omitted.

Convergence

A sequence $(a_{n})_{n \geq 0}$ is said to have a limit $L$ if, for every $ϵ > 0$ , there exists an $N > 0$ such that

| a_{n} - L | < ϵ for all n \geq N .

In this case, we say the sequence $(a_{n})$ converges to $L$ and we write

lim_{n \to \infty} a_{n} = L, or (a_{n}) \to L as n \to \infty .

A sequence which does not converge is called divergent.

Algebraic Limit Theorem

The following identities assume $(a_{n})$ and $(b_{n})$ are convergent sequences such that

lim_{n \to + \infty} a_{n} = A and lim_{n \to + \infty} b_{n} = B .

$lim_{n \to + \infty} [(a_{n}) + (b_{n})] = A + B$
$lim_{n \to + \infty} [(a_{n}) - (b_{n})] = A - B$
$lim_{n \to + \infty} [(a_{n}) \cdot (b_{n})] = A \cdot B$
If $B \neq 0$ , $lim_{n \to + \infty} [(a_{n}) \div (b_{n})] = A \div B$

Monotone Convergence

A monotonic sequence is convergent if and only if it is bounded.

An increasing sequence $(a_{n})$ is sometimes denoted as $(a_{n}) ↗$ . A decreasing sequence $(b_{n})$ is sometimes denoted as $(b_{n}) ↘$ .

Asymptotic Equality

Two sequences $(a_{n})$ and $(b_{n})$ of complex numbers are said to be asymptotically equal if

lim_{n \to + \infty} \frac{a_{n}}{b_{n}} = 1.

This relation is denoted as $a_{n} \sim b_{n}$ as $n \to + \infty$ .

Limit Functions

Let $(f_{n})$ denote a sequence of real- or complex-valued functions having a common domain on the real line or in the complex plane. Let $S$ denote the set of points $x$ for which sequence $(f_{n} (x))$ converges. The function $f$ defined on $S$ by the equation

f (x) = lim_{n \to \infty} f_{n} (x) if x \in S,

is called the limit function of the sequence $(f_{n})$ . We say the sequence $(f_{n})$ converges pointwise to $f$ on the set $S$ .

Uniform Convergence

A sequence of functions $(f_{n})$ converges uniformly to $f$ on a set $S$ if for every $ϵ > 0$ there is an $n_{0} \in N$ such that $n \geq n_{0}$ implies

| f_{n} (x) - f (x) | < ϵ for all x in S .

We denote this symbolically by writing " $f_{n} \to f$ uniformly on $S$ ."

Continuity

Assume $f_{n} \to f$ uniformly on an interval $S$ . If each function $f_{n}$ is continuous at a point $p$ in $S$ , then the limit function $f$ is also continuous at $p$ .

Integration

Assume $f_{n} \to f$ uniformly on an interval $[a, b]$ , and assume each $f_{n}$ is Riemann integrable on $[a, b]$ . Define a new sequence $(g_{n})$ by the equation

g_{n} (x) = \int_{a}^{x} f_{n} (t) d t if x \in [a, b]

Then $f$ is Riemann integrable and $g_{n} \to \int_{a}^{x} f (t) d t$ uniformly on $[a, b]$ . That is,

lim_{n \to \infty} \int_{a}^{x} f_{n} (t) d t = \int_{a}^{x} lim_{n \to \infty} f_{n} (t) d t .

Differentiation

Unlike integration, differentiability is not transmitted by uniform convergence. One notable exception to this rule is within a power series's circle of convergence.

Special Cases

∆ᵏ-constant Sequence

Let $(a_{n})$ be a sequence. We define the $k$ th differences of $(a_{n})$ recursively:

The $0$ th differences of $(a_{n})$ is $(a_{n})$ .
The $k$ th differences of $(a_{n})$ is the sequence given by subtracting consecutive terms of the $(k - 1) s t$ differences of $(a_{n})$ .

A sequence is said to be $Δ^{k}$ -constant if the $k$ th differences are constant. The closed formula for a sequence will be a degree $k$ polynomial if and only if the sequence is $Δ^{k}$ -constant.

This is the discrete analog to (continuous) derivatives of polynomials.

Arithmetic

An arithmetic sequence $(a_{n})_{n \geq 0}^{d}$ is a sequence in which each term differs by the common difference constant $d$ .

Closed Formula. $a_{n} = a + n d$ .
Recursive Definition. $a_{n} = a_{n - 1} + d$ given initial term $a_{0}$ .

Geometric

A geometric sequence $(a_{n})_{n \geq 0}^{r}$ is a sequence in which each term differs by a common ratio $r$ .

Closed Formula. $a_{n} = a r^{n}$ .
Recursive Definition. $a_{n} = r a_{n - 1}$ given initial term $a_{0}$ .

Square Numbers

The $n$ th term of the square numbers $(s_{n})_{n \geq 0}$ is $n^{2}$ . The first few terms are

0, 1, 4, 9, 16, 25, 36, 49, 64, \dots

Triangular Numbers

The $n$ th term of the triangular numbers $(T_{n})_{n \geq 0}$ is the sum of whole numbers $\sum_{k = 0}^{n} k$ . The first few terms are

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \dots