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A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, the continued fraction is finite or infinite .
Euler derived the formula as connecting a finite sum of products with a finite continued fraction. (+ (+ (+))) = + + + + = + + + +The identity is easily established by induction on n, and is therefore applicable in the limit: if the expression on the left is extended to represent a convergent infinite series, the expression on the right can also be extended to represent a convergent infinite ...
This definition is helpful in descriptive set theory to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than (hence a set) – and gives a rapid proof of one direction of the Lebesgue integrability condition.
By considering the complete quotients of periodic continued fractions, Euler was able to prove that if x is a regular periodic continued fraction, then x is a quadratic irrational number. The proof is straightforward. From the fraction itself, one can construct the quadratic equation with integral coefficients that x must satisfy.
Continued fractions are, in some ways, more "mathematically natural" representations of a real number than other representations such as decimal representations, and they have several desirable properties: The continued fraction representation for a real number is finite if and only if it is a rational number.
The Rogers–Ramanujan continued fraction is a continued fraction discovered by Rogers (1894) and independently by Srinivasa Ramanujan, and closely related to the Rogers–Ramanujan identities. It can be evaluated explicitly for a broad class of values of its argument.
Here’s what the letters represent: A is the amount of money in your account. P is your principal balance you invested. R is the annual interest rate expressed as a decimal. N is the number of ...
Cumulative distribution function for the exponential distribution Cumulative distribution function for the normal distribution. In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable, or just distribution function of , evaluated at , is the probability that will take a value less than or equal to .