same relation holding true for . Substituting the value of and in () we obtain: which is true because of our induction hypothesis. Specifically, if neither nor is zero () we can express the difference between the th and th convergents like this: The equivalence transformation If is any infinite sequence of non-zero complex numbers we can prove, by induction, that where equality is understood as equivalence, which is to say that the successive convergents of the continued fraction on the left are exactly the same as the convergents of the fraction on the right. The equivalence transformation is perfectly general, but two particular cases deserve special mention. First, if none of the are zero a sequence can be chosen to make each partial numerator a 1: where , , , and in general . Second, if none of the partial denominators are zero we can use a similar procedure to choose another sequence to make each partial denominator a 1: where and otherwise . These two special cases of the equivalence transformation are enormously useful when the general convergence problem is analyzed. Notions of convergence As mentioned in the introduction, the continued fraction converges if the sequence of convergents } tends to a finite limit. This notion of convergence is very natural, but it is sometimes too restrictive. It is therefore useful to introduce the notion of general convergence of a continued fraction. Roughly speaking, this consists in replacing the part of the fraction by , instead of by 0, to compute the convergents. The convergents thus obtained are called modified convergents. We say that the continued fraction converges generally if there exists a sequence such that the sequence of modified convergents converges for all sufficiently distinct from . The sequence is then called an exceptional sequence for the continued fraction. See Chapter 2 of for a rigorous definition. There also exists a notion of absolute convergence for continued fractions, which is based on the notion of absolute convergence of a series: a continued fraction is said to be absolutely convergent when the series where are the convergents of the continued fraction, converges absolutely. The Śleszyński–Pringsheim theorem provides a sufficient condition for absolute convergence. Finally, a continued fraction of one or more complex variables is uniformly convergent in an open neighborhood when its convergents converge uniformly on ; that is, when for every there exists such that for all