Sequences 1

What is a sequence of real numbers?

A function a: ℕ → ℝ.

What do a_n and the index n represent in a sequence?

a_n = a(n) is the term at index n; unless otherwise stated, n belongs to ℕ = {0, 1, 2, 3, ...}.

What is a recursive definition of a sequence?

A definition that specifies one or more initial values and a recursive rule that determines later terms from earlier terms.

What is an explicit definition of a sequence?

An analytical formula that gives the value of any term directly from its index.

What is an arithmetic sequence?

A sequence with a constant first difference: a_n - a_(n-1) = d for every n ≥ 1, where d is the common difference.

What are the recursive and explicit definitions of an arithmetic sequence with a_0 = a and common difference d?

Recursive: a_0 = a and a_n = a_(n-1) + d. Explicit: a_n = a + dn.

What is a geometric sequence?

A sequence with a constant ratio: a_n/a_(n-1) = q for every n ≥ 1, where q is the common ratio.

What are the recursive and explicit definitions of a geometric sequence with a_0 = a and common ratio q?

Recursive: a_0 = a and a_n = q·a_(n-1). Explicit: a_n = aq^n.

What is a piecewise-defined sequence?

A sequence that follows different rules on different parts of its domain.

When is a sequence bounded above?

When there exists M ∈ ℝ such that a_n ≤ M for every n ∈ ℕ. The number M is called an upper bound.

When is a sequence unbounded above?

When, for every M ∈ ℝ, there exists n ∈ ℕ such that a_n > M.

What is the supremum of a sequence?

Its least upper bound: sup a_n = min{M ∈ ℝ : M ≥ a_n for every n ∈ ℕ}.

When is a sequence bounded below?

When there exists m ∈ ℝ such that a_n ≥ m for every n ∈ ℕ. The number m is called a lower bound.

When is a sequence unbounded below?

When, for every m ∈ ℝ, there exists n ∈ ℕ such that a_n < m.

What is the infimum of a sequence?

Its greatest lower bound: inf a_n = max{l ∈ ℝ : l ≤ a_n for every n ∈ ℕ}.

When is a sequence bounded?

When it is bounded both above and below; equivalently, there exists k > 0 such that |a_n| < k for every n ∈ ℕ.

What is an unbounded sequence?

A sequence that is not bounded.

When is a sequence increasing or strictly increasing?

Increasing if a_(n+1) ≥ a_n for every n ∈ ℕ; strictly increasing if a_(n+1) > a_n for every n ∈ ℕ.

When is a sequence decreasing or strictly decreasing?

Decreasing if a_(n+1) ≤ a_n for every n ∈ ℕ; strictly decreasing if a_(n+1) < a_n for every n ∈ ℕ.

What is a constant sequence?

A sequence that is both increasing and decreasing, so a_n = k for every n ∈ ℕ.

What is a monotonic sequence?

A sequence that is either increasing or decreasing.

What is a strictly monotonic sequence?

A sequence that is either strictly increasing or strictly decreasing.

When does a sequence satisfy a property P eventually?

When there exists an index n̄ = n̄(P) such that every term from that index onward satisfies P.

What is the epsilon definition of convergence of a sequence to L ∈ ℝ?

a_n converges to L if, for every ε > 0, there exists n_ε ∈ ℕ such that n ≥ n_ε implies |a_n - L| < ε. The number L is called the limit.

What is the topological definition of convergence of a sequence to L ∈ ℝ?

a_n converges to L if, for every neighborhood B_ε(L) of L, there exists n_ε such that n ≥ n_ε implies a_n ∈ B_ε(L).

What are the common notations for a sequence converging to L?

a_n → L as n → +∞, or lim_(n→+∞) a_n = L.

What equivalent inequalities express |a_n - L| < ε?

L - ε < a_n < L + ε; equivalently, d(a_n,L) < ε.

What proposition characterizes convergence using distance?

A sequence a_n converges to L ∈ ℝ if and only if d(a_n,L) → 0 as n → +∞.

If a_n converges to L, what is L in relation to the sequence?

L is an accumulation point, or limit point, of the sequence; it does not have to be one of the sequence's terms.

When does a sequence converge to L from above?

When lim_(n→+∞) a_n = L and a_n ≥ L for every n, or at least eventually. This is written lim_(n→+∞) a_n = L⁺.

When does a sequence converge to L from below?

When lim_(n→+∞) a_n = L and a_n ≤ L for every n, or at least eventually. This is written lim_(n→+∞) a_n = L⁻.

What is an infinitesimal sequence?

A sequence a_n such that lim_(n→+∞) a_n = 0. It may be written a_n = o(1).

When does a sequence diverge positively?

When, for every K ∈ ℝ, there exists n_K such that n ≥ n_K implies a_n > K. This is written a_n → +∞.

When does a sequence diverge negatively?

When, for every k ∈ ℝ, there exists n_k such that n ≥ n_k implies a_n < k. This is written a_n → -∞.

What is an infinite sequence in the terminology of these slides?

A sequence whose limit as n → +∞ is either +∞ or -∞.

What is a regular sequence?

A sequence that converges in the extended real line ℝ̄; that is, it converges to a finite L ∈ ℝ, diverges positively to +∞, or diverges negatively to -∞.

What is the neighborhood definition of convergence in the extended real line ℝ̄?

A sequence a_n converges to L ∈ ℝ̄ if, for every neighborhood U(L), there exists n_U such that n ≥ n_U implies a_n ∈ U(L).

What is an irregular sequence?

A sequence that is neither convergent to a finite real number nor divergent to +∞ or -∞.