of that convergent and the next convergent. For example, the fraction is greater than , but − is less than = (in fact, − is just more than = ). The demonstration of the foregoing properties is deduced from the fact that if we seek the difference between one of the convergent fractions and the next adjacent to it we shall obtain a fraction of which the numerator is always unity and the denominator the product of the two denominators. Thus the difference between and is , in excess; between and , , in deficit; between and , , in excess; and so on. The result being, that by employing this series of differences we can express in another and very simple manner the fractions with which we are here concerned, by means of a second series of fractions of which the numerators are all unity and the denominators successively be the product of every two adjacent denominators. Instead of the fractions written above, we have thus the series: + − + − ... The first term, as we see, is the first fraction; the first and second together give the second fraction, ; the first, the second and the third give the third fraction , and so on with the rest; the result being that the series entire is equivalent to the original value. Generalized continued fraction A generalized continued fraction is an expression of the form where the an (n > 0) are the partial numerators, the bn are the partial denominators, and the leading term b0 is called the integer part of the continued fraction. To illustrate the use of generalized continued fractions, consider the following example. The sequence of partial denominators of the simple continued fraction of does not show any obvious pattern: or However, several generalized continued fractions for have a perfectly regular structure, such as: The first two of these are special cases of the arctangent function with = 4 arctan (1) and the fourth and fifth one can be derived using the Wallis product. The continued fraction of above consisting of cubes uses the Nilakantha series and an exploit from Leonhard Euler. Other continued fraction expansions Periodic continued fractions The numbers with periodic continued fraction expansion are precisely the irrational solutions of quadratic equations with rational coefficients; rational solutions have finite continued fraction expansions as previously stated. The simplest examples are the golden