On the other hand the class of positive integers is

Info icon This preview shows pages 147–149. Sign up to view the full content.

View Full Document Right Arrow Icon
On the other hand the class of positive integers is not finite but infinite. This may be expressed more precisely as follows. If n is any positive integer, such as 1000, 1 , 000 , 000 or any number we like to think of, then there are more than n positive integers. Thus, if the number we think of is 1 , 000 , 000, there are obviously at least 1 , 000 , 001 positive integers. Similarly the class of rational numbers, or of real numbers, is infinite. It is convenient to express this by saying that there are an infinite number of positive integers, or rational numbers, or real numbers. But the reader must be careful always to remember that by saying this we mean simply that the class in question has not a finite number of members such as 1000 or 1 , 000 , 000. 53. Properties possessed by a function of n for large values of n . We may now return to the ‘functions of n ’ which we were discussing in §§ 50 51 . They have many points of difference from the functions of x which we discussed in Chap. II . But there is one fundamental character- istic which the two classes of functions have in common: the values of the variable for which they are defined form an infinite class . It is this fact which forms the basis of all the considerations which follow and which, as we shall see in the next chapter, apply, mutatis mutandis , to functions of x as well. Suppose that φ ( n ) is any function of n , and that P is any property which φ ( n ) may or may not have, such as that of being a positive integer
Image of page 147

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
[IV : 53] LIMITS OF FUNCTIONS OF A 132 or of being greater than 1. Consider, for each of the values n = 1, 2, 3, . . . , whether φ ( n ) has the property P or not. Then there are three possibilities:— ( a ) φ ( n ) may have the property P for all values of n , or for all values of n except a finite number N of such values: ( b ) φ ( n ) may have the property for no values of n , or only for a finite number N of such values: ( c ) neither ( a ) nor ( b ) may be true. If ( b ) is true, the values of n for which φ ( n ) has the property form a finite class. If ( a ) is true, the values of n for which φ ( n ) has not the property form a finite class. In the third case neither class is finite. Let us consider some particular cases. (1) Let φ ( n ) = n , and let P be the property of being a positive integer. Then φ ( n ) has the property P for all values of n . If on the other hand P denotes the property of being a positive integer greater than or equal to 1000, then φ ( n ) has the property for all values of n except a finite number of values of n , viz. 1, 2, 3, . . . , 999. In either of these cases ( a ) is true. (2) If φ ( n ) = n , and P is the property of being less than 1000, then ( b ) is true. (3) If φ ( n ) = n , and P is the property of being odd, then ( c ) is true. For φ ( n ) is odd if n is odd and even if n is even, and both the odd and the even values of n form an infinite class. Example. Consider, in each of the following cases, whether ( a ), ( b ), or ( c ) is true: (i) φ ( n ) = n , P being the property of being a perfect square, (ii) φ ( n ) = p n , where p n denotes the n th prime number, P being the property of being odd, (iii) φ ( n ) = p n , P being the property of being even, (iv) φ ( n ) = p n , P being the property φ ( n ) > n , (v) φ ( n
Image of page 148
Image of page 149
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern