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.

LIMITS

7

by the binomial theorem, and
— ( I ) Q I )( 2)
+i_ _-i-[ - -j -I
I --—... I - ---
(I + -+ I + ( ZIn + I) + I n + I/ I
4... ++
K i+, +1~1~ + +
J il + I
by changing mz to in + I. Whence it is plain that
I+ > I+,
71l q- I V"^
since, after two terms, each term of + is greater than
the corresponding term of I +, and, moreover, there is
one more term in I + >. Hence, zhclnt x is increased
by Zunity, x beinzg any positive iztcger, +I - increases.
Again, it is clear that I +-) can never be greater than
the series
I+I+,I-+-+***+ =I++1.
12 3 1 1X
This last series is less than
I+I+ f,f~~~+ 2 22-1 21' 2111-1
Whence < 3 - I
n being a positive integer. If we take in to be a very large
positive integer, we see that I + <3 however large In
may be, since it is always less than a quantity which can never
be greater than 3. Therefore, since ( +-) increases continually when z does, z being a positive integer, but at the same
time is always less than 3, and since +) = 2, we see
\ / _z==