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.

SUMMAT ION

3

z=5
log z^- log 6o.
(- I)-' log^ _ logo:. ii 5 6 7 8 9 10
1* 10
X( i)_-=- X(- i )Z I _ I I I +_ I I +1 I
the form +(<,) + (,,n + I) +... + (to find the equivalent+).
/It
exprsin - + ). co s n exa le te t- sins 2 sin. \ 2 4 \2 6in 2 24
6 9 2547
4. Ino e the preceding pararaph e ere conc erned with find1 1
thain the value of hich); in the present and following article
wfactor of the sthe inverse problem, avingr given a series of
the foterm hence(t)+ 3+ i 2 ( in the th) to ermfind the eis, uivalent
express ion (f). As an example take the seri es
Wae obse ter tveate at the signs of the even numbered
terms are negative; this is effected in the summation by introducing the sig'z facfor (- I):-1, so that when,v is odd the sign
is +. We notice further that each nlumerator is greater by 2
than the number of the term in which it stands, while the first
factor of the denominator is 3 greater. The last factor of
the denominator increases 2 units per term, and is 5 in the
first term, hence it is 3 + 2 71 in the nzth term; that is, it exceeds
twice the number of the term by 3. The nth term, therefore,
expressed as a function of n, may be written
(_,),,_, <, + 2)
(I - + 3)(2,Z + 3)
and, all together, there are 22 terms.. 3 4 +.. 524 = (- I)'l-(X + 2')
"4'5 5.7 25' 47 = (x+ 3)(2r+ 3)