ALGEBRA 25 



(a + nh) (a -f n + m ih) 



+ _A_( I 1 



(m i)h ( a(a + h) . . . (a + m 2h) (a + nh) . . . (a + n + m 2h) J 

 + . + 



h (a a + nh 



1.856 If f(x) is a polynomial of degree m it can be expressed : 



/(*) = Ao + Ai(x + mti) + A 2 (x + mh)(x + m- ih) + . . . . 



+ A m (x + mh) . . . (x + h) 

 and, 



a+nh 



f(x) _ = A, ( 



x(x + h) . . . (x + mh) mh \ a(a + h) .... (a + m iti) 



\ 



n ih J 



(a + nh) (a + m + n ih) 



a+nh 



where, 



a+nh 



x a a + h a + 2h a + n 



a 



1.86 Euler's Summation Formula. 



+ .... + A m - 



x=b 



/A V^ 



*-to2t 

 x = a 



d m f(x + h-z) 



m!0(z), with & = i, is the Bernoullian polynomial. 



^! = i, y! 2fc + 1 = o; the coefficients ^ 2 fc are connected with Bernoulli's 

 numbers (6.902), B k , by the relation, 



II I 



2' Z '~ 12' 720' 



