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/ |
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