Tract 9.

all their terms positive.

20S

very same theorem as before, the sum s of the original serieswill then be expressed thus, s =

a — bx — -

a - x

b'x — (c'-

b' z

a!

(cl'-

2 b'c' b' 1

+ ‘

&c;

a: 1 a’

where the series in the last denominator, having again thesame properties with the former one, will have its first twoterms very large in respect of the following terms. But thesefirst two terms, a'—b'x, are themselves very small in compa-rison with the first two, a ~ bx, of the former series; andtherefore much more are the third, fourth, &c, terms of thislast denominator, very small in comparison with the samea—bx: for which reason they may now perhaps he smallenough to be neglected.

A. But if these last terms he still thought too large to heomitted, then find the sum of them by the very same theorem :and thus proceed, by repeating the operation in the samemanner, till the required degree of accuracy is obtained.Which it is evident, will happen after a small number of re-petitions, because that, in each new denominator, the third,fourth, &c, terms, are commonly depressed, in the scale ofnumbers, two or three places lower than the first and secondterms are. And the general theorem, denoting the sum swhen the process is continually repeated, will he this,

Ml

a —bx — -

a a xx

a’ — bx — —a"

a a xx

■ b"x ■

a a xx

a'" — O’".v ■

a" 1 a" 1 xx

- b“"x See.

6 - But the general denominator d in the fraction —, which

is assumed for the value of s or a + bx + ex* + &c, maybe otherwise found as below ; from which the general law of