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Section 8.1: Sequences. Definition. A sequence is a function whose domain is ℕ. My Informal Definition. A sequence is a list . Notation. or sometimes. Sequences. s. Sequences. s. Example. ( 1/n ) converges to 0 Idea: When n gets big, 1/n gets small. Tower of Power.
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Definition A sequence is a function whose domain is ℕ.
My Informal Definition A sequence is a list.
Notation or sometimes
Example ( 1/n ) converges to 0 Idea: When n gets big, 1/n gets small
Examples Converges to 3
Examples Diverges to infinity
Examples Converges to 0
Examples Converges to 0
Examples Diverges to ∞
Theorem Suppose (sn) converges to s and (tn) converges to t.
Squeeze Theorem Then Suppose and
Definitions is increasing if for all n is strictlyincreasing if for all n is decreasing if for all n is strictlyincreasing 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.
Notice that we have k’s raised to the kth power. This is a job for L’Hopital’s Rule!
First, we need to commit algebra. We change the problem by taking the log.
Again, we need to commit algebra. We change the problem by taking the log.
