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Internet Resources for the preCalculus Student

Sequences and Series

A sequence is just a list of numbers. If the list goes on forever then it may be called an infinite sequence. One example is the counting numbers themselves. Another might be their squares: 1, 4, 9, 16, .... (a string of dots is often used to indicate that the sequence is intended to go on "forever", and if the pattern is not obvious it may be indicated by giving a formula for the general term like this "1, 4, 9, ... , n^2, ...").

Two important special classes of sequences are the "arithmetic" sequences whose successive terms differ by the same amount (eg 2, 5, 8, 11, .. , 2+3n, ...), and the "geometric" sequences for which successive terms are related by the same factor (eg 2, 6, 18, 54, .. , 2*3^n, ... ).

One thing you should be aware of is that although you might have guessed the rule that was being used to generate the sequence 1, 4, 9, 16, ... above, there are in fact many other rules which might have given the same first few numbers, so questions which ask you to predict how a sequence will continue generally don't have a unique answer unless some restriction is given on the kinds of rules which can be used. The On-Line Encyclopedia of Integer Sequences  produces many possible rules for continuing whatever you give it as the first few numbers.

But we can also consider sequences whose members are not necessarily integers. For example 1, 1/2, 1/3, ... , 1/n, ...   In this case the numbers in the sequence get closer and closer to zero and so we say thet the sequence "converges to zero" or "has limit zero".
Another example with a limit is the sequence 0.9, 0.99, 0.999, 0.9999, ... (can you see that in this case the limit is 1?) The idea of a "limit" occurs in many applications (for example we might want to predict the eventual behaviour of a system over an infinitely long time period, or to consider a family of better and better approximations to a result that we can't find exactly), and it is fundamental to many of the concepts of calculus.
For more about limits you might check out the Limits and Continuity links in the Calculus section of this guide.

Sequences of numbers often arise in applications (eg daily temperatures, annual harvests of a crop, etc) and frequently there is a rule or formula which gives a good prediction of each value from previous ones (such as when population or bank balance grows by an amount proportional to its current size). Such rules are often called Recurrence Relations. Here are a couple of examples

Recurrence(1)@ies  ,  Recurrence(2)@ies
Such systems are often studied in a Finite or Discrete Math course.


The word "series" in common language implies much the same thing as "sequence", but in mathematics when we talk of a series we are referring in particular to sums of terms in a sequence (eg for a sequence of values a(n) the corresponding series is the sequence of s(n) with s(n) = a(1) + a(2) + .... + a(n-1) + a(n) ).

The relationship between these is shown pictorially in an applet ( Progression of Differences@ies ) from the IES group in Japan.

If the terms are in an arithmetic sequence we call the sums an arithmetic series, and for a geometric sequence we call the sums a geometric series.

For an arithmetic series, one way to establish a formula for the sum is to take the terms in pairs (first+last, second+next to last, and so on), or to take two copies and match them "head to tail". Can you see that this gives 1+2+3+...+n = n(n+1)/2 ?

And for a geometric series we can use an algebraic trick to show that
1 + r + r^2 + ... +r^n = (1- r^(n+1))/(1-r) = 1/(1-r) - r^(n+1)/(1-r)

Formulas for series are often also proved by Mathematical Induction.

For an infinite sequence of a(n), if the values get small fast enough then the sums might have a limit. In that case we say that the infinite series converges.

This is true for example for a geometric series if the ratio is less than one in absolute value.(Since for 1 + r + r^2 + ... +r^n = (1- r^(n+1))/(1-r) = 1/(1-r) - r^(n+1)/(1-r),
 if |r| < 1 then large powers of r will get very small and the second term will go to zero as n goes to infinity, so that the infinite sum is just 1/(1-r).)

This discussion of geometric series (from Frank Wattenberg at Montana State University) includes many interesting practical applications
and has been reviewed by Tony Wang

A study of infinite series is often included in the 2nd semester of a Calculus course.

You might also check our 'raw list' (of links provided without comment) to see if there are any more examples there that we haven't yet included here.

If you have come across any other good web-based illustrations of these and related concepts, please do let us know and we will add them here.

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