Introduction to Sequences
Convergent, Divergent and Oscillating Sequences
Everyone knows a sequence when they see one. People standing in a queue form a sequence. A list of tasks that someone writes out to plan their day is also a sequence.
A sequence is an arrangement of objects, people, tasks, grocery items, books, movies, or numbers, which has an ‘order’ associated with it. To have an order associated with it, it must be in correspondence with the set of natural numbers. More rigorously, a sequence is a function from the set of natural numbers to any given set. This set may be finite or infinite. And although elements may not be repeated in a set, they may be repeated in a sequence.
The set of natural numbers itself forms a sequence. 1, 2, 3, 4, 5… is a sequence. So is 1, 1, 2, 2, 3, 3…, even though the numbers are repeating.
A formula can be used to conveniently describe a sequence without listing out all its terms. For example, if t(n) is the nth term of a sequence, then t(n) = 1/n describes a sequence.
Sequences with Limits – Convergent Sequences
A sequence, by definition, has infinitely many terms, because there is an nth term for every natural number n, and there are infinitely many natural numbers. What happens to the terms of a sequence as n grows in magnitude?
Consider a train track. Its two edges appear to ‘meet’ in the distance, far away from the observer. Each bar between the two edges of the track seems progressively shorter, with increasing distance from the observation point. The bars seem to disappear after a point, and the edges seem to merge, even though they never actually do.
For a sequence such as t(n) = 1/n, the terms of the sequence become smaller as n increases, and they move closer and closer to zero. Even though there is no value of n for which the nth term actually is zero, given any small number d, a number n can be found such that | t(n) – 0 | < d.
To state it more generally, a sequence t(n) is said to converge to a limit l, if, for any given number d, however small, there exists a natural number n such that | t(n) – l | < d.
Divergent Sequences
Not all sequences have limits. Consider t(n) = n. It just gets bigger and bigger with each succeeding term. It cannot possibly converge to anything.
A sequence t(n) is said to diverge in the positive direction, if, for any given number R, however large, there exists a natural number n such that t(n) > R. It is said to diverge in the negative direction, if, for any given negative number R, however large in magnitude, there exists a natural number n such that t(n) < R.
Oscillating Sequences
Not all sequences are quite as well behaved as convergent or divergent sequences. Some sequences do neither of those things. For instance, the sequence 1, -1, 1, -1, 1… does not converge or diverge. It just oscillates.
An oscillating sequence may be bounded or unbounded. The sequence above is bounded, because it oscillates between the bounds [-1, 1]. A sequence like -1, 2, -3, 4, -5… is not convergent, divergent, or bounded. It is an unbounded oscillating sequence.
Suggested Further Reading
Introduction to Real Analysis, Robert G. Bartle & Donald R. Sherbert (Third Edition: Wiley, John & Sons, 1999)
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