1 1 n 1 n p being the property φ n 1 vi φ n 1 1 n 1

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) = 1 - ( - 1) n (1 /n ), P being the property φ ( n ) < 1, (vi) φ ( n ) = 1 - ( - 1) n (1 /n ), P being the property φ ( n ) < 2, (vii) φ ( n ) = 1000 { 1 + ( - 1) n } /n , P being the property φ ( n ) < 1,
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[IV : 54] POSITIVE INTEGRAL VARIABLE 133 (viii) φ ( n ) = 1 /n , P being the property φ ( n ) < . 001, (ix) φ ( n ) = ( - 1) n /n , P being the property | φ ( n ) | < . 001, (x) φ ( n ) = 10 , 000 /n , or ( - 1) n 10 , 000 /n , P being either of the properties φ ( n ) < . 001 or | φ ( n ) | < . 001, (xi) φ ( n ) = ( n - 1) / ( n + 1), P being the property 1 - φ ( n ) < . 0001. 54. Let us now suppose that φ ( n ) and P are such that the asser- tion ( a ) is true, i.e. that φ ( n ) has the property P , if not for all values of n , at any rate for all values of n except a finite number N of such values. We may denote these exceptional values by n 1 , n 2 , . . . , n N . There is of course no reason why these N values should be the first N values 1, 2, . . . , N , though, as the preceding examples show, this is frequently the case in practice. But whether this is so or not we know that φ ( n ) has the property P if n > n N . Thus the n th prime is odd if n > 2, n = 2 being the only exception to the statement; and 1 /n < . 001 if n > 1000, the first 1000 values of n being the exceptions; and 1000 { 1 + ( - 1) n } /n < 1 if n > 2000, the exceptional values being 2, 4, 6, . . . , 2000. That is to say, in each of these cases the property is possessed for all values of n from a definite value onwards . We shall frequently express this by saying that φ ( n ) has the property for large , or very large , or all sufficiently large values of n . Thus when we say that φ ( n ) has the property P (which will as a rule be a property expressed by some relation of inequality) for large values of n , what we mean is that we can determine some definite number, n 0 say, such that φ ( n ) has the property for all values of n greater than or equal to n 0 . This number n 0 , in the examples considered above, may be taken to be any number greater than n N , the greatest of the exceptional numbers: it is most natural to take it to be n N + 1.
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[IV : 55] LIMITS OF FUNCTIONS OF A 134 Thus we may say that ‘all large primes are odd’, or that ‘1 /n is less than . 001 for large values of n ’. And the reader must make himself familiar with the use of the word large in statements of this kind. Large is in fact a word which, standing by itself, has no more absolute meaning in mathematics than in the language of common life. It is a truism that in common life a number which is large in one connection is small in another; 6 goals is a large score in a football match, but 6 runs is not a large score in a cricket match; and 400 runs is a large score, but £ 400 is not a large income: and so of course in mathematics large generally means large enough , and what is large enough for one purpose may not be large enough for another. We know now what is meant by the assertion ‘ φ ( n ) has the property P for large values of n ’. It is with assertions of this kind that we shall be concerned throughout this chapter.
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