Sequences Explained in 5 Minutes

Sequences Explained in 5 Minutes

In mathematics a sequence is simply a list of objects. What kind of objects? Any kind, but they are often numbers. For example, {0,1,1,0} is a sequence of 4 numbers. It turns out that things get more interesting if you have an infinite sequence. For example {1,2,3,...} is the infinite sequence of integers.

Notice that the objects in a sequence are ordered, and objects can repeat. These two properties make sequences very different than sets.

Sequences can "store" a lot of information. For example, even using just two objects [0,1] it's possible to make an infinite number of different sequences!

Another important property of infinite sequences is "convergence". Consider this sequence of fractions..

1/1,1/2,1/3,1/4,1/5,...

The general element of this sequence is (1/n) where n is an integer that goes from 1 to infinity. Notice that as n gets bigger the elements of the sequence get smaller and approach 0. They never reach 0, but they get closer and closer. When this happens mathematicians say that the sequence converges to the limit of 0. Convergence is an important property of infinite sequences.

A sequence whose terms become arbitrarily close together as n gets bigger is called a Cauchy sequence, named for the French mathematician Baron Augustin-Louis Cauchy. The above sequence is an example of a Cauchy sequence. Cauchy sequences are important in several areas of mathematics, probably because of the fact that a sequence is convergent if and only if it is Cauchy.

Sometimes the terms of a sequence are specified by a rule that depends on previous terms. These rules are called "recurrence relations". Here's an example..

Suppose I give you the first two terms of a sequence 0,1 and the rule "the next term of the sequence is the sum of the two prior terms". By repetition of this simple rule you can generate the entire sequence..

0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,...

This is the famous Fibonacci sequence. It was first mentioned by the Indian mathematician Pingala in about 250 BC and has been studied by mathematicians ever since!

Another important class of infinite sequences are the "oscillating" sequences. These don't converge, but they don't diverge either! Here's a couple of examples..

0,1,0,1,0,1,0,1,0...
0,1,2,3,2,1,0,-1,-2,-3,-2,-1,0,1,2,3,2,1,0,-1,-2,-3,-2,-1,0..

The second example clearly shows the "wave like" nature of oscillating sequences. They have a "wavelength" and "amplitude".

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Content written and posted by Ken Abbott abbottsystems@gmail.com
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