1, 2] NUMBER. 3 



exponents of the negative powers increase regularly in absolute 

 value, from a certain term on the coefficients are all obtained by 

 the repetition of a certain definite group : 



a = a n b n + a n _! b n ~ l + ... + a 



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e.g. ~ = 3-10 + 2-10- 1 , 



o 



~ = HO 1 + 010 + HO- 1 + 610- 2 + 610- 3 + 

 o 



No irrational number can be so expressed, though by taking a 

 sufficient number of terms we may obtain a number differing from 

 the given irrational by as little as we please. The coefficients in 

 this case never repeat indefinitely. Since irrationals can not be 

 expressed by means of a finite number of terms each of which is 

 rational, they ate defined by their properties, or as the limit 

 approached by an infinite sequence of rational numbers. 



2. Limits. If we have a sequence of rational numbers, 

 a x> a 2 , a 3 , ..., following each other according to a given law, and 

 can find a number A, possessing the property that, corresponding 

 to any arbitrarily given positive number e however small, we can 

 find a number //, such that for all values of n greater than /A, 



\a n -A <e, n>fj,, 



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then A is called the limit of the sequence. 

 E.g. the sequence 



=1 a -1 - a -1 l - -1 - - - 



has the limit 2, a rational number. 



The necessary and sufficient condition for the existence of a 

 limit is that when e is arbitrarily given, we can find a number /*, 

 such that for all integral values of n greater than //,, and for any 

 positive integral value of p, 



If the latter condition is fulfilled, even though the sequence 

 has no rational limit, the sequence has a limit, which defines an 

 irrational number. 



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