Sequences And Series Flashcards
What is a monotone sequence?
A sequence {xn} of real numbers is said to be monotone for all n ∈ N if:
xn ≤ xn+1 or xn + 1 ≤ xn
What is a monotone increasing sequence?
xn ≤ xn+1
What is a monotone decreasing sequence?
xn+1 ≤ xn
When is a subset of real numbers said to be bounded below?
If there exists an @ ∈ R such that:
@ ≤ x for all x that is an element in the subset S.
When is a subset of real numbers said to be bounded above?
If there exists an ß ∈ R such that:
x ≤ ß for all x that is an element in the subset S.
How do you tell when a monotone sequence converges or not?
An increasing monotone sequence bounded above converges, and a decreasing monotone sequence bounded below also converges.
How do you tell if a function is increasing or decreasing?
If the derivative of the function is negative, then it is decreasing; if the derivative of the function is positive, then it is increasing.
