204

OF SLOW SERIES WITH

TRACT 9

summed, in comparison with the original one, because all thecoefficients after the second term are evidently very small;and indeed they are so much the smaller, and litter for sum-.jnation, by how much the coefficients of the original seriesare nearer to equality; so that, when these a, b, c, d, &c, arequite equal, then the third, fourth, &c, coefficients of the newseries become equal to nothing, and the sum accurately equal

to—— r~ — —~—=—— ; which is also known to he true

a— bx a—ax 1 — x

from other principles,

9, Though the first two terms, a — bx, of the new series,bo very great in comparison with each of the following terms,yet these latter may not always be small enough to be entirelyrejected when much accuracy is required in the summation.And in such case it will be necessary to collect a great num-ber of them, to obtain their sum pretty near the truth ; be-cause their rate of converging is but small, it being indeedpretty much like to the rate of the original series, but onlythe terms, each to each, are much smaller, and that commonlyjn a degree to the hundredth or thousandth part.

4, The resemblance of this new series however, beginningwith the third term, to the original one, in the law of pro-gression, intimates to us that it will be best to sum it in th®Very same manner as the former. Hence then putting

c

a

b' = d

, 2 bd + c * s b-c b 4

c = t — --- i , —~.

a ^ a 1 a 1 *

&c,

and consequently the proposed series a + bx + ex’ 1 + &e,

— bx — a'x 2 — b'x 3 — cx*ikc a — bx — x l xbj’ taking the sum of the scries a' -j- b'x -f- c'x~ -(- &c, by th®