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.

38

SUMMATION OF DIFFERENCES

31. From the foregoing examples we gain some idea of the
utility and importance of the notion of a limit of a sum. The
following problems will illustrate further:
EXERCISES
1. Let the student solve the example of the area of the curve -o X
by finding the sums in terms of ax instead of n, using an identity
which he should be able to form for himself.
2. Solve the example of the falling body by the method of the example of area. In this form it involves an arithmetical progression.
3 Lt _ ( + 1) X (6 X+ 4 XIl + ~Z2 )h2
3. Ax 'o xAx — =? Use xA= 4 4x.4.
a+x z
where Ax _ h, letting p be a quantity which lies between the greatest
and least of the n terms of X 6 x2 + 4 xh + /'.
4
b 3 b4 ' /4
Anzs. f xbdx = ---.
4 4
4. Find fX.
r~dx Lt Ax Lt zA^ Lt
SOLUTION. J b= Lt X X LtX - Ax t b
xa Ax _ o Y Xa Ax -- o _x: Ax x"_
a+Ax a aS4Ax
bd t..t
where T and S are any finite numbers; thus -- AX Lt X O.
x X xI
Ax l (3 x + 2 aX) AxAlso +
3 2 X- 2 (X + AX)'' 2 X:'(X + AX)2
fbdx Lt b x Lt 2 T
X3 2 d"- 2( a, + x A) Xp j
where p and Axnp =(b + Ax - a)p are finite quantities, n being the
number of terms in X(3x 2 AX). Thus Ax2p vanishes when
AX =( O. X )2

Therefore

rdx I I
J. X~ " 2 2 b2