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TRACT IX.

A METHOD OF SUMMING THB SERIES a + bx + CX l + dx 3 -j-ex* + Kc, WHEN IT CONVERGES VERY SLOWLY, NAMELY,WHEN X IS NEARLY EQUAL TO 1, AND THE COEFFICIENTS

a, b, c, d, Kc, decrease very slowly : the signs of

ALL THE TERMS BEING POSITIVE.

ARTICLE I.

When there is occasion to find the sum of such series asthat above-mentioned, having the coefficients a, b, c, d, &c,of the terms, decreasing very slowly, and the convergingquantity x pretty large ; the sum cannot be found by col-lecting the terms together, on account of the immense num-ber of them which it would be necessary to collect; neithercan it be summed by means of the differential series, because

X

the powers of the quantity -—-will then diverge faster than

the differential coefficients converge. In such case then wemust have recourse to some other method of transforming itinto another series which shall converge faster. The follow-ing is a method by which the proposed series is changed intoanother, which converges so much the quicker as the originalseries is slower.

2. The method is thus. Assume — = the given series

d be =

a + bx + cx % -f- dx 3 + Ac. Then shalla 1

a+6jr+cjr I -l-&c

b 1 C 2bc b 3

-{c--)s?-(d - ~+-)x 3 -(e-

; which, by actual division, is=a — bx

2bd.-\-c l 3b*c b 4

a ' 1 a ‘ a“ ' a + ~~

See. Consequently a z divided by this series will be equal tothe series proposed ; and this new series will be very easily