"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 "MATHEMATICS: The Loss of Certainty" by Morris Kline.
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.
(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.
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).
But if the ratio is bigger than 1 the terms get bigger so the successive sums get further apart rather than closer together. What happens when the ratio is exactly equal to -1?
Assuming that the sum formula works when the ratio has a magnitude that is not less than 1 can lead to paradoxes like a "proof" that -1=infinity.
Other series can often be tested for convergence by comparing them with geometric series.
An "infinite degree polynomial" is generally called a Power Series. This looks like a geometric series but won't actually be one unless the coefficients are all equal (or are themselves a geometric sequence). Power series which arise as limiting cases of Taylor Polynomials are called Taylor Series.
And there is an lab
on Taylor Polynomials and Taylor Series that is part of UBC's CalculusOnline
program. (You won't be able to "hand in" your answers, but will still be
able to read the material and do all the activities)
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
please do let us know and we will add them here.
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