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.

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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.

 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 an < bn < cn for all n greater than some positive integer N, then Example.

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

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 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.