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.

I8

SUMMATION OF DIFFERENCES

that I + - must approach some limiting value which lies
between 2 and 3. Call this limit e, and the theorem is proved
for a positive integer.
To prove the theorem for a positive fractional value of x.
Let 0 be a positive fraction, and Iz and n two consecutive
integers, such that
Iz =;-1+ I, and > 0 > z.

Then
and

(I +-) <( + ) <(I + --
(I +-) < (I + 0 < (I+

~/ I?z-a / IY Y ^n+/3
that is, (I +- <(+ < I +-
a and / being positive proper fractions; hence
(~ + -- )?~z/I\<
(I + )
If 0 is made to increase without limit, rz- and?n consequently
do the same,
(I +- -)=I (II+ >)=I, (i+>I) c, and I +;
therefore (I + ) e, 0 being a positive fraction, and the
theorem is proved for positive fractional values of x.
To prove the theorem for ncgative values of x. Let x be
any positive number, and put y = - x; hence y is numerically
equal to x, but negative. We have
( + i Y - ( I - ) x.( 7 _ ( ) (+

i.e.

I + =I X I \-.
I + - -=-I + -- I +... --- -.
)y}/ x - I x - I)