Understanding Monotonic Sequences: Boundedness and Convergence Properties
This text discusses monotonic sequences, focusing on non-decreasing and non-increasing sequences. It explains how a non-decreasing sequence (a_n) that is bounded above converges to a limit (a) less than or equal to the upper bound (A), while a non-increasing sequence (b_n) that is bounded below converges to a limit (b) greater than or equal to the lower bound (B). Additionally, it covers how to prove monotonicity using algebraic differences, ratios, and derivatives, with examples illustrating each method.
Understanding Monotonic Sequences: Boundedness and Convergence Properties
E N D
Presentation Transcript
Theorem Suppose {an} is non-decreasing and bounded above by a number A. Then {an} converges to some finite limit a, with a A. Suppose {bn} is non-increasing and bounded below by a number B. Then {bn} converges to some finite limit b, with b B.
Boundedness • A sequence is bounded provided there is a positive number K such that |an| Kfor all values of n.
Monotonicity • A sequence {an} is monotone if either • anan+1 or • anan+1 for all n. • The sequence is non-decreasing if situation a holds, and non-increasing if situation b holds.
Proving Monotonicity: Part 1 • Algebraic Difference:1) If an+1– an 0 for all n, then the sequence is non-increasing.2) If an+1– an 0 for all n, then the sequence is non-decreasing. • ExampleShow that the sequence is monotone.
Proving Monotonicity: Part 2 • Algebraic Ratio: If an> 0 for all n, then1) If an+1/an 1 for all n, the sequence is non-increasing.2) If an+1/an 1 for all n, the sequence is non-decreasing. • ExampleShow that the sequence is monotone.
Proving Monotonicity: Part 3 • Derivative: If an = f (n), then1) If f(n) 0 for all n, the sequence is non-increasing.2) If f(n) 0 for all n, the sequence is non-decreasing. • ExampleShow that the sequence is monotone.
