[3; 7, 15, 1] = [3; 7, 16] and thus is a convergent of any number strictly between {| cellpadding="2" cellspacing="0" | align="right" | |||| |- | align="right" | |||| |} Comparison Consider and . If is the smallest index for which is unequal to then if and otherwise. If there is no such , but one expansion is shorter than the other, say and with for , then if is even and if is odd. Continued fraction expansion of and its convergents To calculate the convergents of we may set , define and , and , . Continuing like this, one can determine the infinite continued fraction of as [3;7,15,1,292,1,1,...] . The fourth convergent of is [3;7,15,1] = = 3.14159292035..., sometimes called Milü, which is fairly close to the true value of . Let us suppose that the quotients found are, as above, [3;7,15,1]. The following is a rule by which we can write down at once the convergent fractions which result from these quotients without developing the continued fraction. The first quotient, supposed divided by unity, will give the first fraction, which will be too small, namely, . Then, multiplying the numerator and denominator of this fraction by the second quotient and adding unity to the numerator, we shall have the second fraction, , which will be too large. Multiplying in like manner the numerator and denominator of this fraction by the third quotient, and adding to the numerator the numerator of the preceding fraction, and to the denominator the denominator of the preceding fraction, we shall have the third fraction, which will be too small. Thus, the third quotient being 15, we have for our numerator , and for our denominator, . The third convergent, therefore, is . We proceed in the same manner for the fourth convergent. The fourth quotient being 1, we say 333 times 1 is 333, and this plus 22, the numerator of the fraction preceding, is 355; similarly, 106 times 1 is 106, and this plus 7 is 113. In this manner, by employing the four quotients [3;7,15,1], we obtain the four fractions: , , , , .... To sum up, the pattern is These convergents are alternately smaller and larger than the true value of , and approach nearer and nearer to . The difference between a given convergent and is less than the reciprocal of the product of the denominators