- #1
Playdo
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What is the value of the simple continued fraction [1;2,3,5,7,11,13,...,nth prime] as n goes to infinity?
What is the value of this infinite continued fraction?
- Thread starter Playdo
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In summary, a continued fraction is a mathematical expression in the form of a fraction where the numerator and denominator are both integers, and the denominator is a sum of an integer and a fraction. It is different from a regular fraction in that the denominator is more complex and can represent irrational numbers. Continued fractions have many applications in mathematics, physics, and engineering, and their value can be found using different methods such as using a calculator or following an algorithm. They can also be infinite, and their value can be approximated to any desired degree of accuracy.
- #2
g_edgar
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You can compute it to as many decimals as you like. There is absolutely no reason to think this constant can be written in any simpler way.
- #3
The Chaz
- 206
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g_edgar said:
...but if you find a pattern, you'd better run it by my before telling anyone else!
Related to What is the value of this infinite continued fraction?
1. What is a continued fraction?
A continued fraction is a mathematical expression in the form of a fraction where the numerator and denominator are both integers, and the denominator is a sum of an integer and a fraction. It is denoted by [a0; a1, a2, a3, ...] and is used to represent a real number.
2. How is a continued fraction different from a regular fraction?
A regular fraction is written as a/b, where a and b are both integers. In a continued fraction, the denominator is a sum of an integer and a fraction, making it a more complex representation of a fraction. Continued fractions are also used to represent irrational numbers, while regular fractions can only represent rational numbers.
3. What are some applications of continued fractions?
Continued fractions have many applications in mathematics, physics, and engineering. They are used to approximate real numbers and to solve equations. They also have applications in number theory, cryptography, and signal processing.
4. How do you find the value of a continued fraction?
To find the value of a continued fraction, you can either use a calculator or follow the algorithm for evaluating continued fractions. This involves finding the convergents of the continued fraction and taking the limit as the number of terms approaches infinity. Alternatively, you can use the continued fraction as a recursive algorithm to approximate the value.
5. Can continued fractions be infinite?
Yes, continued fractions can be infinite. This means that the continued fraction has an infinite number of terms, and the value of the continued fraction cannot be determined exactly. However, the value of the continued fraction can be approximated to any desired degree of accuracy by taking the limit as the number of terms approaches infinity.
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