An introduction to the summation of differences of a function; an elementary exposition of the nature of the algebraic processes replaced by the abbreviations of the infinitesimal calculus, by B. F. Groat.

AREAS AND LIMITS OF SUMS

33

31 11 21 31
'. _= +-+ '=6.4355
2 2 12 22
30 31
and = — yal/ +y/ll = 6.1655.
1 2
Hence, 6.4355 >A >6.i655, and the error made in taking
either number for A is less than 0.27 units; shown also by
(Y31 -l)l =- 0.27.
Of course it is easy to see now, from (y,,+ -yl)hi, that for 300
subdivisions of (b -a) the error would be less than 0.027; for
3000 subdivisions less than 0.0027; and so on. Let us see if the
limit can be found.
26. Introducing the literal values given in the example, we
have
b- Ax x=b
-:Xa-X2 <A4< 1( AX 2AX;;r= —f c+a+x
that is,
a + (a +( + h)2+. /. +(a -- /) <A,
A, < a+/i+(a +21/)+...-+(a+n/i
A naH/- + / ~12 A,
b-1L 2+ b-12 2 )(7i( J-n)2(sz-I)12(2 -I)] A
IO iZ 1 ].
I0 0 ii
b-a b -a n(n+ I) (b-a)' l(z I)(2 n- jI).
Whence
b-a +(-)( ( 2 I
a (b-a)I -,- + (b - ay <A,
b-n I OL 2I7
b-a a 2 + a~>A.
a 2 _ (b-a)(KI + a(b- ) >6
1o _ 6 >