3.3.3 THEORETICAL (Web hosting servers) TESTS 77 Fig. 7. The sawtooth
3.3.3 THEORETICAL TESTS 77 Fig. 7. The sawtooth function ((z)), The proof of Theorem P indicates that a priori tests can indeed be carried out, provided that we are able to deal satisfactorily with sums involving the 1 J and [ 1 functions. In many cases the most powerful technique for dealing with floor and ceiling functions is to replace them by two somewhat more symmetrical ones: if z is an integer; 6(z) = [ZJ + 1 - 121 = 
1 f if z is not an integer; (6) ((z)) = 2 - [ZJ - 4 + &Y(z) = z - [zl + 3 - g?(z). (7) The latter function is a sawtooth function familiar in the study of Fourier series; its graph is shown in Fig. 7. The reason for choosing to work with ((z)) rather than [z] or [zl is that ((z)) p assesses several very useful properties: K-z)) = -((z)); (8) ((2 + n)) = ((z)), integer n; (9) Nnz)) = C(z))+ ((z + k)) + . . . + ((2 + G)), integer n 2 1. 00) (See exercise 2.) In order to get some practice working with these functions, let us prove Theorem P again, this time without relying on exercise 1.2.4-37. With the help of Eqs. (7), (8), (9), we can show that I2- s(x) 1 x- s(x) = -(( —(x )) + f -+(x-;(x)) m m = x - s(x) -x-c;f >>+; m cc = x -4×1 +((qq)+; (11) m since (2 - s(z))/ m is never an integer. Now x - 4×1 = 0 c m O