Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

BOOK Two
In the geometrical fractional series +- + 4 + i +
1
A+++ f + L
1. 27
++a+&+
&c.
&c.
++*+&+& &c.
the last term multiplied by the sum of the denominators is
equal to the sum of the series, when the number of terms is
finite.
Thus $X 1+2+4+8=? the sum of 1 + $ + + + 4. Thus
&X 1+3+9+27=%, and so of the rest. And this will be
the case whatever the ratio of the series may be.
And in the first of these series if unity be taken from the denominator
of the last term, and the remainder be added to the
denominator, the sum arising from this addition multiplied by
the last term will be equal to the sum of the series. Thus 8-
1=7 and 7+8=15, and +~15=?, the sum of the series.
In the second of these series, if unity be subtracted from the
denominator of the last term, the remainder be divided by 2,
and the quotient be added to the said denominator, the sum
multiplied by the last term will be equal to the sum of the
series. In the 3rd series after the subtraction the remainder
must be divided by 3: in the 4th by 4: in the fifth by 5: in the
6th by 6, and so on.
In the series 1 + + + + & + &, &c. which infinitely continued
is equal to 2, an infinitesimal excepted, if any finite
number of terms is assured, then if the denominator of the last
term be added to the denominator of the term immediately
preceding it, and the sum of the two be multiplied by the last
term, the product will be equal to the sum of the series. Thus
in two terms 1+3~*=$= the sum of l+a. If there arc
three terms, then 3 + 6 X % = = 1 + $ + -& If there are four
terms, then 6 + 10 X -1-u-
rest.
10 - 10 - 1 +++ti-&,
and so of the