Multivariable Chapter 12.1

Front 1

Infinite Sequence

Back 1

a function whose domain is the set of integers greater than or equal to the first term

Front 2

Monotonic Sequence

Back 2

a sequence that is either increasing or decreasing

Front 3

Precise definition of convergence

Back 3

The sequence {an} CONVERGES to the number L if for every epsilon greater than 1, there exists an integer N, such that the absolute value of an - L is less than epsilon, for all n>N.

If no such number L exists, we say that {an} DIVERGES

In {an} converges to L, we write lim {an} as n aproaches infinity = L.

Front 4

Upper bounded sequence

Back 4

there exists a number M such that an is less than or equal to M, for all n. The number M is called an upper bound for {an}. If M is the minimum of all upper bounds of {an}, then M is called the LEAST UPPER BOUND of {an}.

Front 5

Lower Bound Sequence

Back 5

there exists a number m such that an is greater than or equal to m, for all n. The number m is called a lower bound for {an}. If m is the maximum of all lower bounds of {an}, then m is called the GREATEST LOWER BOUND of {an}.

Front 6

Monotonic Sequence Theorem

Back 6

Every bounded monotonic sequence is convergent.

A monotonic increasing sequence that is not bounded above is divergent.

A monotonic decreasing sequence that is not bounded below is divergent.

Front 7

Squeeze Theorem

Back 7

If {an} </= {bn} </= {cn} for n >/= and lim {an} as n approaches infinity = lim {cn} as n approaches infinity = L, then lim {bn} as n approaches infinity = L