Definition. A sequence is a function whose domain is the set of all positive integers.

If a is a sequence then we write an = a(n) for each positive integer n.

Examples.

Using the TI-85 graphing calculator to plot sequences.
Discussion [Using Flash] [Using Java]

Definition. The limit of a sequence {an} is L, written

if, for each e > 0, there is an integer N such that | L - an | < e whenever n > N.

If the limit of the sequence {an} exists then we say the sequence {an} is convergent; otherwise, we say the sequence {an} is divergent.

Examples.

Some properties of limits.

Corollary 1. If {an} and {bn} are sequences with the property that there exists some integer N such that an = bn for all n > N then {an} is convergent if and only if {bn} is convergent.

If the sequences are convergent then their limits are the same.

Theorem. If there exists a function f which is defined for all real numbers x > 1 such that an = f(n) for all positive integers n and such that the limit

exists then

Examples.

Corollary 2. Consider the sequence {an} for some real number a. If -1 < a < 1 then {an} is convergent; otherwise, {an} is divergent.

If -1 < a < 1 then

if a = 1 then

Corollary 3. Consider the sequence {np} for some real number p. If p < 0 then {np} is convergent; otherwise, {np} is divergent.

If p < 0 then

if p = 0 then

Squeeze Theorem. Suppose that {an}, {bn} and {cn} are sequences such that
  1. an < bn < cn for all n greater than some positive integer N,
then

Example.

Corollary 4. If {an} is a sequence such that

then

Discussion.

Example.

Monotone Convergence Theorem. Suppose that {an} is a sequence such that, for some real number B, either
  • an < an+1 < B for all positive integers n
or
  • an > an+1 > B for all positive integers n.
Then the sequence {an} is convergent.

Example.