APPENDIX. 
Summation of Series by Approximative Fractions. 
By R. J. Apcocx. 
THs method of summation of series consists in putting the 
series into a continued fraction by common algebraic division, 
and then finding its approximating fractions until one is found 
which gives the sum of the series either approximately or 
entirely. 
Hx. Find the sum of the series a + ar + ar? + ar*+ &e. 
a+ar+ar*+ar+&c.) 1 fF 
l+rtr+r+é&e. Q 
—r—r—r—&e.) a+ar+ar+ar+ &e.(—“ 
a+ar+ar+ar+&e,. ” 
no remainder. 
Then 
oe aor tke = a+ar+ar+ar+ ee ee 
reat ae 
a 
¢ 
which is the entire sum of the series and the exact algebraic 
expression from which it is derived. To find the sum of n 
terms, it is to be observed that beginning with the (n+1)th 
= the series is ar*"+ar"*'+ar"*+ &c.=by the same process, 
= Hence a+ ar+tarrtarFrt+....ar= 
: Ait the common formula for the sum of x 
l-r 1l-r 1—r ’ 
terms of a geometrical series. 
The superiority of this method of summation over others is: 
First, its comparative simplicity, on account of which it is 
worthy of a place in common algebra. 
Second, it is more generally applicable than any method _ 
known to me. ; 
Third, the facility with which it gives the entire sum of a 
series when capable of being expressed in finite terms, and the 
rapidity with which it approximates to the sum of those capa- 
ble of being expressed only approximately. 
