Sequences and Series - Core Definitions

Sequences and series extend calculus to discrete settings, providing tools for approximation, convergence analysis, and representation of functions. Understanding limits of sequences is fundamental to infinite series and power series.

DefinitionSequence

A sequence is a function whose domain is the set of positive integers. We denote the sequence as $\{a_n\}_{n=1}^\infty$ or simply $\{a_n\}$ , where $a_n$ is the $n$ -th term.

Examples:

$\{n^2\} = 1, 4, 9, 16, 25, \ldots$
$\left\{\frac{1}{n}\right\} = 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots$
$\{(-1)^n\} = -1, 1, -1, 1, -1, \ldots$

DefinitionLimit of a Sequence

We say that $\{a_n\}$ converges to $L$ , written $\lim_{n \to \infty} a_n = L$ , if for every $\epsilon > 0$ , there exists $N$ such that $n > N \implies |a_n - L| < \epsilon$

If no such $L$ exists, the sequence diverges.

ExampleConvergent Sequences

$\lim_{n \to \infty} \frac{1}{n} = 0$ : The terms get arbitrarily close to 0
$\lim_{n \to \infty} \frac{n+1}{n} = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right) = 1$
$\lim_{n \to \infty} \frac{n^2}{2n^2 + 1} = \lim_{n \to \infty} \frac{1}{2 + 1/n^2} = \frac{1}{2}$

ExampleDivergent Sequences

$\{n\}$ diverges to infinity
$\{(-1)^n\}$ oscillates between $-1$ and $1$ , so has no limit
$\{(-1)^n n\}$ oscillates with unbounded magnitude

TheoremLimit Laws for Sequences

If $\lim_{n \to \infty} a_n = A$ and $\lim_{n \to \infty} b_n = B$ , then:

$\lim_{n \to \infty} (a_n + b_n) = A + B$
$\lim_{n \to \infty} (ca_n) = cA$ for any constant $c$
$\lim_{n \to \infty} (a_n b_n) = AB$
$\lim_{n \to \infty} \frac{a_n}{b_n} = \frac{A}{B}$ if $B \neq 0$
$\lim_{n \to \infty} (a_n)^p = A^p$ if $p > 0$ and $a_n > 0$

TheoremSqueeze Theorem for Sequences

If $a_n \leq b_n \leq c_n$ for all $n$ beyond some point, and $\lim_{n \to \infty} a_n = \lim_{n \to \infty} c_n = L$ , then $\lim_{n \to \infty} b_n = L$

ExampleUsing the Squeeze Theorem

Find $\lim_{n \to \infty} \frac{\sin n}{n}$ .

Since $-1 \leq \sin n \leq 1$ for all $n$ : $-\frac{1}{n} \leq \frac{\sin n}{n} \leq \frac{1}{n}$

Since $\lim_{n \to \infty} \frac{1}{n} = 0$ and $\lim_{n \to \infty} -\frac{1}{n} = 0$ , by the Squeeze Theorem: $\lim_{n \to \infty} \frac{\sin n}{n} = 0$

DefinitionMonotonic Sequences

A sequence $\{a_n\}$ is:

Increasing if $a_n \leq a_{n+1}$ for all $n$
Decreasing if $a_n \geq a_{n+1}$ for all $n$
Monotonic if it is either increasing or decreasing

TheoremMonotone Convergence Theorem

Every bounded monotonic sequence converges.

Specifically:

If $\{a_n\}$ is increasing and bounded above, it converges to $\sup\{a_n\}$
If $\{a_n\}$ is decreasing and bounded below, it converges to $\inf\{a_n\}$

ExampleApplication of Monotone Convergence

Consider $a_n = \left(1 + \frac{1}{n}\right)^n$ .

This sequence is increasing and bounded above by 3 (can be shown using binomial theorem). By the Monotone Convergence Theorem, it converges. The limit is the famous constant: $\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e \approx 2.71828$

Remark

Connection to functions: If $f(x)$ is defined for $x \geq 1$ and $a_n = f(n)$ , then $\lim_{n \to \infty} a_n = \lim_{x \to \infty} f(x)$ when both limits exist. This allows us to use L'Hôpital's Rule and other continuous function techniques for sequences.

Sequences provide the foundation for understanding series, which are infinite sums of sequence terms.