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It is sometimes necessary to separate a continued fraction into its even and odd parts. For example, if the continued fraction diverges by oscillation between two distinct limit points p and q, then the sequence {x 0, x 2, x 4, ...} must converge to one of these, and {x 1, x 3, x 5, ...} must converge to the other.
In the analytic theory of continued fractions, Euler's continued fraction formula is an identity connecting a certain very general infinite series with an infinite continued fraction. First published in 1748, it was at first regarded as a simple identity connecting a finite sum with a finite continued fraction in such a way that the extension ...
If x is rational, it will have two continued fraction representations that are finite, x 1 and x 2, and similarly a rational y will have two representations, y 1 and y 2. The coefficients beyond the last in any of these representations should be interpreted as +∞; and the best rational will be one of z(x 1, y 1), z(x 1, y 2), z(x 2, y 1), or ...
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.
The rate of convergence depends on the absolute value of the ratio between the two roots: the farther that ratio is from unity, the more quickly the continued fraction converges. When the monic quadratic equation with real coefficients is of the form x 2 = c, the general solution described above is useless because division by zero is not well ...
The continued fractions on the right hand side will converge uniformly on any closed and bounded set that contains no poles of this function. [ 7 ] In the case 2 F 1 {\displaystyle {}_{2}F_{1}} , the radius of convergence of the series is 1 and the fraction on the left hand side is a meromorphic function within this circle.
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An extreme example: if a set X is given the discrete topology (in which every subset is open), all functions : to any topological space T are continuous. On the other hand, if X is equipped with the indiscrete topology (in which the only open subsets are the empty set and X) and the space T set is at least T 0, then the only continuous ...