SEQUENCE, CONVERGENCE, LIMIT, ACCUMULATION POINT, BOUND

Def. Sequence. A sequence is a set of numbers, quantities or elements arranged in a definite order.

Examples of sequences.

1.         2, 4, 6, 8, 10, ..., 2n, ... .

3.         {x, 2x², 3x³, ... , nxⁿ}

6.         1, 0, 1, 0, 1, 0, ...

7.         any progression (arithmetic, geometric, harmonic)

8.         the terms of a series

9.         the partial sums of a series

A sequence can be either finite or infinite. An infinite sequence is non-terminating, there being another term after each term. A finite sequence has only a definite number of terms. The i-th term of a sequence is often denoted by a(i) or a_i. Sequences are denoted by notations such as {a₁, a₂, a₃, .... , a_n, .... }, {a_n}, (a_n). A sequence can be viewed as a particular kind of function, a function whose independent variable n ranges over the set of positive integers.

Q. The elements of a sequence can be what? We speak of sequences of numbers (integers, rational numbers, real numbers, etc.). Can the elements of a sequence be anything besides numbers?

A. They can also be

• complex numbers

• points in the plane

• points in 3-space

• points in n-space

• functions

Limit of a sequence. A sequence of numbers {s₁, s₂, s₃, ... , s_n, ... } has the limit s if, for any prescribed accuracy, there is a position in the sequence such that all terms after this position approximate s within this prescribed accuracy i.e for any ε > 0 there exists an N such that |s - s_n| < ε for all n greater than N. A series of points {p₁, p₂, p₃, ... } has the limit p if, for each neighborhood U of p there is a number N such that p_n is in U if n > N.

Example. The sequence

has a limit of 0. The n-th term approaches 0 and meets the requirement of the definition.

                                                                                                James and James. Mathematics Dictionary

Theorem. If a sequence has a limit, the limit is unique.

Def. Convergent sequence. A sequence that has a limit is said to be convergent. Otherwise, it is said to be divergent.

Def. Accumulation point (or cluster point or limit point) of a sequence. A point P such that there are an infinite number of terms of the sequence in any neighborhood of P.

Example. The sequence

has two accumulation points, the numbers 0 and 1.

Bound to a sequence. An upper bound to a sequence of real numbers is a number which is equal to or greater than every number in the sequence. A lower bound to a sequence of real numbers is a number which is equal to or less than every number in the sequence. If a sequence has both an upper bound and a lower bound, it is said to be a bounded sequence. The smallest upper bound is called the least upper bound. The largest lower bound is called the greatest lower bound.

                                                                                                James and James. Mathematics Dictionary 

Limit superior. For a sequence of real numbers, the largest accumulation point is called the limit superior and denoted by lim sup or . A number is called the limit superior if infinitely many terms of the sequence are greater than - ε for any positive ε, while only a finite number of terms are greater than + ε.

Syn. greatest limit, maximum limit, upper limit

Limit inferior. For a sequence of real numbers, the smallest accumulation point is called the limit inferior and denoted by lim inf or . A number is called the limit inferior if infinitely many terms of the sequence are less than + ε for any positive ε, while only a finite number of terms are less than - ε.

Syn. least limit, minimum limit, lower limit

Cauchy’s condition for convergence of a sequence. An infinite sequence converges if, and only if, the numerical difference between every two of its terms is as small as desired, provided both terms are sufficiently far out in the sequence. Tech. The infinite sequence s₁, s₂, s₃, ... , s_n, ... converges if, and only if, for every ε > 0 there exists an N such

            | s_n+h - s_n | < ε

that for all n > N and all h > 0.

                                                                                                James and James. Mathematics Dictionary

Def. Cauchy sequence. A Cauchy sequence is a sequence where, given any preassigned positive number ε, however small, there exists a point in the sequence (possibly very far out) beyond which the distance between any two selected elements is less than ε. Tech. A Cauchy sequence is a sequence of points P₁, P₂, ... such that for any ε > 0 there is a number N for which ρ(P_i, P_j) < ε if i > N and j > N, where ρ(P_i, P_j) is the distance between P_i and P_j. If the points are points of Euclidean space, this is equivalent to the sequence being convergent. If the points are real (or complex) numbers, then ρ(P_i, P_j) is | P_i - P_j | and the sequence is convergent if and only if it is a Cauchy sequence.

Syn. Convergent sequence, fundamental sequence, regular sequence.

                                                                                                James and James. Mathematics Dictionary

Monotonic sequences.

Monotonic increasing sequence. A sequence of real terms a₁, a₂, ... ,a_n, ... such that a_n+1 a_n for all n i.e. a sequence in which the terms either increase or remain the same.

Monotonic decreasing sequence. A sequence of real terms a₁, a₂, ... ,a_n, ... such that a_n+1 a_n for all n i.e. a sequence in which the terms either decrease or remain the same.

Theorem. Every monotonic (increasing or decreasing) sequence a₁, a₂, ... ,a_n, ... with the property that |a_n| < M (a constant) converges. That is, every bounded monotonic sequence has a limit.

Theorems on limits of sequences. For sequences consisting of numbers the following hold:

References.

   James and James. Mathematics Dictionary.

   Taylor. Advanced Calculus.

   Spiegel. Advanced Calculus. (Schaum)