Sequences

Informal description

Digits of $π$

3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, \dots

Approximations of $\frac{1}{3}$

0.3, 0.33, 0.333, 0.3333, 0.33333, \dots

Recursive

Fibonacci numbers

\begin{matrix} f_{0} = 0, & f_{1} = 1, & f_{n} = f_{n - 1} + f_{n - 2} \\ 0, 1, 1, 2, 3, 5, 8, \dots \end{matrix}

First order recurrence

\begin{matrix} a_{0} = -1, & a_{n} = 4 + a_{n - 1} \\ -1, 3, 7, 11, 15, \dots \end{matrix}

Algebraic description

{\frac{n}{n + 1}}_{n = 0}^{\infty} = 0, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \dots

{\frac{n + 2}{n + 3}}_{n = 0}^{\infty} = \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \dots

{\frac{n}{n + 1}}_{n = 2}^{\infty} = \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \dots

Special sequences

Arithmetic

\begin{matrix} {4 n - 1}_{n = 0}^{\infty} & = & -1, 3, 7, 11, 15, \dots \\ {n}_{n = 0}^{\infty} & = & 0, 1, 2, 3, 4, \dots \end{matrix}

Geometric

\begin{matrix} {3 {(\frac{1}{2})}^{n}}_{n = 0}^{\infty} & = & 3, \frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \frac{3}{16}, \dots \\ {{(-1)}^{n}}_{n = 0}^{\infty} & = & 1, -1, 1, -1, 1, -1, \dots \end{matrix}

Sequences

Terminology

A sequence in $ℝ$ is a function $𝒶 : ℕ \to ℝ$ . The value

𝒶 (i) = 𝒶_{i}

is called the term of the sequence with index $i$ . A sequence

𝒶_{0}, 𝒶_{1}, 𝒶_{2}, 𝒶_{3}, \dots

is denoted as ${𝒶_{n}}_{n = 0}^{\infty}$ or simply ${𝒶_{n}}$ .

Arithmetic sequence

A sequence of the form

{α n + β}_{n = 0}^{\infty}

is called arithmetic sequence.

Geometric sequence

A sequence of the form

{γ 𝓇^{n}}_{n = 0}^{\infty}

is called geometric sequence.

Sequences

Limits

Limit

Let ${𝒶_{n}}$ be a sequence then $𝒶 \in ℝ$ is its limit

\lim_{n \to \infty} 𝒶_{n} = 𝒶 {𝒶_{n}} \to 𝒶

\forall ε \in ℝ^{+}, \exists N (ε) \in ℕ : N (ε) \leq m \Rightarrow | 𝒶 - 𝒶_{m} | < ε

If the limit exists the sequence converges, else it diverges.

Properties

Let ${𝒶_{n}} \to 𝒶$ and ${𝒷_{n}} \to 𝒷$ . Then

${α 𝒶_{n} + β 𝒷_{n}} \to α 𝒶 + β 𝒷$
${𝒶_{n} 𝒷_{n}} \to 𝒶 𝒷$
${\frac{𝒶_{n}}{𝒷_{n}}} \to \frac{𝒶}{𝒷}$ provided that $𝒷 \neq 0$ and $𝒷_{n} \neq 0$

Computing via functions

Let $f (x)$ be a real valued function defined for all natural numbers $n$ larger than $N_{0}$ . Let ${𝒶_{n}}$ be a sequence such that

f (i) = 𝒶_{i}

for all $N_{0} \leq i$ . Then

\lim_{x \to \infty} f (x) = 𝒶 \Rightarrow {𝒶_{n}} \to 𝒶

Remark: converse is false

{0}_{n = 0}^{\infty} \to 0 \lim_{x \to \infty} (x sin (x π)) = DNE

Continuous functions

Let ${𝒶_{n}} \to 𝒶$ and $f (x)$ be continuous on $𝒶$ . Then $\exists M \in ℕ$ such that

f (𝒶_{n})

is defined for all $n \geq M$ and

{f (𝒶_{n})} \to f (𝒶)

Squeeze theorem

Let ${𝒶_{n}}$ , ${𝒷_{n}}$ and ${𝒸_{n}}$ be sequences. If $\exists M \in ℕ$ such that

M \leq n \Rightarrow 𝒶_{n} \leq 𝒷_{n} \leq 𝒸_{n}

and ${𝒶_{n}} \to ℒ$ , ${𝒸_{n}} \to ℒ$ then

{𝒷_{n}} \to ℒ

Sequences

Examples

\begin{matrix} {0}_{n = 0}^{\infty} & = & 0, 0, 0, 0, 0, 0, \dots \\ {{(-1)}^{n}}_{n = 0}^{\infty} & = & 1, -1, 1, -1, 1, -1, \dots \\ {(1 - {(-1)}^{n}) n}_{n = 0}^{\infty} & = & 0, 2, 0, 4, 0, 6, \dots \\ {\frac{1}{n}}_{n = 1}^{\infty} & = & 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \dots \end{matrix}

\begin{matrix} {5 + \frac{3}{n^{2}}}_{n = 1}^{\infty} & = & 8, \frac{23}{4}, \frac{48}{9}, \frac{83}{16}, \dots \end{matrix}

\begin{matrix} {\frac{2^{n}}{n^{2}}}_{n = 1}^{\infty} & = & 2, 1, \frac{8}{9}, 1, \frac{32}{25}, \frac{16}{9}, \dots \\ {{(1 + \frac{2}{n})}^{n}}_{n = 1}^{\infty} & = & 3, 4, \frac{125}{27}, \frac{81}{16}, \dots \end{matrix}

{cos (\frac{3}{n^{2}})}_{n = 1}^{\infty}

{\frac{{| cos n |}^{n}}{n^{2}}}_{n = 1}^{\infty} {\frac{{(-1)}^{n}}{2^{n}}}_{n = 1}^{\infty}

{\frac{4^{n}}{n!}}_{n = 1}^{\infty}

a_{1} = 2 a_{n + 1} = \frac{a_{n}}{2} + \frac{1}{2 a_{n}}

2, \frac{5}{4}, \frac{41}{40}, \frac{3281}{3280}, \frac{21523361}{21523360}, \dots

Sequences

bounded and monotone

Bounded

above

A sequence ${𝒶_{n}}$ is bounded from above if

\exists 𝓊 \in ℝ : \forall n \in ℕ 𝒶_{n} < 𝓊

$𝓊$ is an upper bound.

Bounded

below

A sequence ${𝒶_{n}}$ is bounded from below if

\exists 𝓁 \in ℝ : \forall n \in ℕ 𝓁 < 𝒶_{n}

$𝓁$ is an lower bound.

Bounded

A sequences is bounded if it is bounded from above and from below.

Monotone

increasing

A sequence ${𝒶_{n}}$ is monotone increasing if

\exists M \in ℕ : M \leq n \Rightarrow 𝒶_{n} \leq 𝒶_{n + 1}

𝒶_{M} \leq 𝒶_{M + 1} \leq 𝒶_{M + 2} \dots

Monotone

decreasing

A sequence ${𝒶_{n}}$ is monotone decreasing if

\exists M \in ℕ : M \leq n \Rightarrow 𝒶_{n} \geq 𝒶_{n + 1}

𝒶_{M} \geq 𝒶_{M + 1} \geq 𝒶_{M + 2} \dots

Monotone

A sequence is monotone if it is monotone increasing or monotone decreasing.

Monotone convergence theorem

Every bounded monotone sequence is convergent.