Math question, concerning continued fractions (convergents)?

Question

I have happened upon a new curiousity... continued fractions... and convergents...

This problem I saw was indicated as: 1/1+... which essentially was a series of 1/(1 + 1) / (1 +1) / 1...) without bound.

My query is: What branch of mathematics is this type of idea studied in? I would like to study more things like this but do not know where to look.

Any help is appreaciated.

Answer 1

I think the idea of convergent fractions is older then Euler.



A similar idea, look up Zeno's Paradox,



http://en.wikipedia.org/wiki/Zeno's_paradoxes



Read the Achilles and the Tortoise, basically it puts the fastest man on earth against a tortoise with a small head start. Achilles never passes because first he must get half way to the turtle, then from that point another half, then from that point another half way to the turtle, and so on. Basically this idea stumped them because they could not determine after how many seconds Achilles (the fastest man on earth) would pass the tortoise. Because it seemed he would only be getting closer.



It was because at this point there was still no concept of 0 (or infinity. Zero and infinity have always been dual objects, it was always taken that you could not have one without the other.) As far as I know as the first example of an infinite converging series of fractions.



Although this is not a continued fraction, it has the same idea and I think is something a little more profound, it is I think the first example of an infinite convergent sequence, I think that questions like this got people curious. And thinking about continuous motion and things going off to infinity and such.



A basic number theory book may have more information on this. I would think a good introduction to analysis may have a bit more on convergent sequences then an algebra book would. However, no books come to my mind, it's not something I have read about; most of my books are more general and abstract this is more specific.



Wikipedia is always a good source (and contains other good sources!) I just scanned it and I think maybe I'll read it later when I have more time but that would probably be a good start. Also look at the sources for the article as often those books are pretty good too.



Actually the first recorded use of continued fractions is 300 BC by Euclid. (see wiki article)

However Zeno was around 400 BC, so the example I gave (while not a continued fraction) is older.



http://en.wikipedia.org/wiki/Continued_fraction



Good luck!

Answer 2

The first mathematician to start systematizing a theory of

continued fractions was L. Euler. I recommend Hardy and

Wright's, Introduction to the Theory of Numbers. You will

also get theorems on approximation because convergents

are the best approximations for a given sized denominator.

This is a little branch of number theory. An acquantaince

is attempting collect all important or interesting theorems

on continued fractions and organize them in a book. I

have to find out how far along he is on this project. He is

a professor at Texas A & M. Update: you can get the book

used i found $70 lowest price so far. It's"Continued Fractions" by Doug Hensley.

Actually, if your starting out, some very cheap number theory books at B&N have a chapter on continued fractions. And

some of these can be had used for under $10.

Happy Converging>

Answer 3

Do a search on:



Number theory

Infinite sequences

and

Infinite series



Should be a good start...



:-)