What is a continued fraction?

Date: 03/06/98 at 20:08:31
From: Jim Wollak
Subject: What is a continued fraction?

Dear Dr. Math,

In one of your archived problems from Oliver Wong on 6/3/96 regarding 
the number "e," you stated that "e" can be expressed as a CONTINUED 
FRACTION.

  http://mathforum.org/dr.math/problems/wong6.3.96.html   

I'm familiar with fractions, but continued ones? Are these fractions 
or functions expressed like a fraction? I'm not too smooth with the 
Net, but I did try to skim your archived problems, dictionary, and 
commonly asked questions about this subject. I'm curious about the 
term, and would simply like to know more about it (In other words, 
it's not for a class.).

My real questions are, what is a continued fraction and what makes it 
different from the types of fractions or ratios I'm used to?

Thank you very much.

Jim W.

Date: 03/06/98 at 23:17:57
From: Doctor Wolf
Subject: Re: What is a continued fraction?

Hi Jim,

As explained in the archived problem you mention, a continued fraction 
is a fraction of the form:


                             1
     X = A_0 + --------------------------------
                                  1
                A_1 + -------------------------
                                     1
                       A_2 + ------------------
                                         1
                              A_3 + -----------
                                     A_4 + ...

These values can often also be expressed as an infinite series. For 
example, e can be calculated to any precision (based on Taylor's 
Theorem) by the series:

    e = 2 + 1/2! + 1/3! + 1/4! + 1/5! + ... forever

where 2! = 1*2, 3! = 1*2*3, 4! = 1*2*3*4, etc.

Since the expansion goes on forever, it is an example of an infinite 
series. I've provided a few others below. It's an interesting exercise 
to evaluate each of these with a calculator and see how close the 
infinite series version is to the actual answer after only a few 
terms or series are considered.

    cos(x) = 1 - x^2/2!   + x^4/4!   - x^6/6! + ... x in radians
    sin(x) = x - x^3/3!  + x^5/5!  - x^7/7! + ...   x in radians
    e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + ... (let x = 1 to find e)

Excellent question.

-Doctors Wolf and TWE, The Math Forum
Check out our web site! http://mathforum.org/dr.math/