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

I9

Let x oo; then
y- 00 (I +- ' e and ( + I-.
Hence the limit, when y approaches negative infinity, of
( I '
+- is e.
Therefore I + )] - e whether x be positive, negative,
integral, or fractional. Q.E.D.
SCHOLIUM. For other proofs of this theorem see Todhunter's
Differential Calculls.
14. The Value of e. The student may be interested to know
the value of e. By definition, c= t ( +-), and this in
whatever way x is made to approach infinity, whether positive,
negative, integral, or fractional. Then let Im be a positive
integer; whence
(- - I + - ) - 2
+ m) [2 I n -
I ---.... I -
(( )
Let the sum of the first In terms of this series be denoted by
S,,, and the remainder after n terms by R,,; that is,
I -...* I — ) (I.. —.. I ----
R('
R,-= + +...,
the remaining (m + I -- u) terms of S,,+i.
Hence < 1- +R1+
Ii n + I + ' 1
</i~ I +-+.+ ** + —)1-
\ 1 i - 7n-\