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### What is a continued fraction?

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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
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.
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```
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/
```
Associated Topics:
High School Calculus
High School Definitions
High School Sequences, Series

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