Section 8.1: Sequences

Definition Informal Definition Notation A sequence is a function whose domain is ℕ. Informal Definition A sequence is a list. Notation

Sequences s

Sequences s

Theorem Suppose (sn ) converges to s and (tn ) converges to t.

Squeeze Theorem Suppose and Then

Tower of Power nn n! 3n en n4 n2 n √n ∜n ln n

Examples Diverges to infinity

Examples Converges to 0

Examples Converges to 0

Examples Diverges to ∞

Important Facts to Know

Definitions is strictly increasing if for all n is increasing if for all n is strictly increasing if for all n is decreasing if for all n is strictly decreasing if for all n is monotone if it is either increasing or decreasing

Definitions is bounded above by M if for all n is bounded below by M if for all n is bounded if it is bounded both above and below

Bounded Monotone Convergence Theorem An increasing sequence converges if and only if it is bounded above. A decreasing sequence converges if and only if it is bounded below. A monotone sequence converges if and only if it is bounded.