Internet Resources for the Calculus Student

Sequences and Series

"In his first paper on the Calculus (1669), Newton proudly introduced the use of infinite series to expedite the processes of the calculus... As Newton, leibnitz, the several Bernoullis, Euler, d'Alembert, Lagrange, and other 18th-century men stuggled with the strange problem of infinite series and employed them in analysis, they perpetuated all sorts of blunders, made false proofs, and drew incorrect conclusions; they even gave arguments that now with hindsight we are obliged to call ludicrous."

>From "MATHEMAICS: The Loss of Certainty" by Morris Kline.

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. Many calculus courses rely on an intuitive understanding of the notion of a limit, and leave the precise formal definition to a later course in "Analysis" (at Langara that would be Math2373).

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 to sums of terms in a sequence.
These  Notes on Series written by prof. Jim Carlson at the University of Utah may help you to see how we can make sense of the sum of an infinite sequence by looking at the sequence of "partial sums".

The sums of geometric sequences are called geometric series, and can be shown to converge whenever the ratio of successive terms has magnitude less than 1.
A nice pictorial demonstration of this (intended to be accessible to quite young children but still maybe useful for adults) is provided by "Mathman" Don Cohen (whose site includes many other examples of how calculus concepts can be explored by young children - which may also help older students see things a new way and avoid "drowning" in the algebra).

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

This discussion of Taylor Series has been reviewed by Sahar Khalili.
 
 
 

Series

Progression of Differences@ies

Recurrence(1)@ies


Recurrence(2)@ies


 
 

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 good web-based illustrations of these and related concepts,
     please do let us know and we will add them here.