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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) = n2

f: ℕ → ℝ, n → f(n) = (-1)n

f: ℕ → ℝ, n → f(n) = 1/n, n ≠ 0

f: ℕ → ℝ, n → f(n) = an ⇔ {an}n∈ℕ

f: ℕ → ℝ, n → f(n) = n2 ⇔ {n2}n∈ℕ = {1,4,9...}

A sequence {an}n∈ℕ satisfies a certain property definitively if ∃ n0 ∈ ℕ | the property is true ∀ n > n0.

{n-100}n∈ℕ is definitively positive, in fact n-100 > 0 ∀ n > 100, where 100 is n0

{(-1)n}n∈ℕ is not definitively positive; ∀ n0 ∃ n > n0, n = 2k+1 | (-1)n = -1 < 0

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

{2n}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: {nα}n∈ℕ, α ∈ ℝ.

{n1/2}n∈ℕ is an harmonic sequence with α = 1/2

{n4}n∈ℕ is an harmonic sequence with α = 4

{1/n2}n∈ℕ is an harmonic sequence with α = -2, 1/n2 = n-2

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

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

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

A sequence {an}n∈ℕ is strictly monotonic decreasing if an > an+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.

{-n2}n∈ℕ is a monotonic decreasing sequence.

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