SEQUENCE

A sequence is a function from ℕ to ℝ.

A sequence is a function that has a domain of natural numbers, and a codomain of real numbers.

f: ℕ → ℝ

n → f(n) ∈ ℝ

f: ℕ → ℝ, n → f(n) = n^{2}

f: ℕ → ℝ, n → f(n) = (-1)^{n}

f: ℕ → ℝ, n → f(n) = 1/n, n ≠ 0

f: ℕ → ℝ, n → f(n) = a_{n} ⇔ {a_{n}}_{n∈ℕ}

f: ℕ → ℝ, n → f(n) = n^{2} ⇔ {n^{2}}_{n∈ℕ} = {1,4,9...}

A sequence {a_{n}}_{n∈ℕ} satisfies a certain property definitively if ∃ n_{0} ∈ ℕ | the property is true ∀ n > n_{0}.

{n-100}_{n∈ℕ} is definitively positive, in fact n-100 > 0 ∀ n > 100, where 100 is n_{0}

{(-1)^{n}}_{n∈ℕ} is not definitively positive; ∀ n_{0} ∃ n > n_{0}, n = 2k+1 | (-1)^{n} = -1 < 0

Geometric sequence or geometric progression: {q^{n}}_{n∈ℕ}, q ∈ ℝ.

{2^{n}}_{n∈ℕ} is a geometric sequence with q = 2

{(1/3)^{n}}_{n∈ℕ} is a geometric sequence with q = 1/3

{(-1)^{n}}_{n∈ℕ} is a geometric sequence with q = -1

Harmonic sequence or harmonic progression: {n^{α}}_{n∈ℕ}, α ∈ ℝ.

{n^{1/2}}_{n∈ℕ} is an harmonic sequence with α = 1/2

{n^{4}}_{n∈ℕ} is an harmonic sequence with α = 4

{1/n^{2}}_{n∈ℕ} is an harmonic sequence with α = -2, 1/n^{2} = n^{-2}

A sequence {a_{n}}_{n∈ℕ} is monotonic increasing if a_{n} ≤ a_{n+1} ∀ n ∈ ℕ

A sequence {a_{n}}_{n∈ℕ} is monotonic decreasing if a_{n} ≥ a_{n+1} ∀ n ∈ ℕ

A sequence {a_{n}}_{n∈ℕ} is strictly monotonic increasing if a_{n} < a_{n+1} ∀ n ∈ ℕ

A sequence {a_{n}}_{n∈ℕ} is strictly monotonic decreasing if a_{n} > a_{n+1} ∀ n ∈ ℕ

{n}_{n∈ℕ} is a monotonic increasing sequence.

{1/n}_{n∈ℕ} is a monotonic decreasing and bounded sequence.

{-1/n}_{n∈ℕ} is a monotonic increasing and bounded sequence.

{√n}_{n∈ℕ} is a monotonic increasing sequence.

{-n^{2}}_{n∈ℕ} is a monotonic decreasing sequence.

{(-1)^{n}}_{n∈ℕ} is not a monotonic sequence.