( H6 )

TRACT VIII.

A NEW METHOD FOR THE VALUATION OF NUMERAL INFI-NITE SERIES, WHOSE TERMS ARE ALTERNATELY (+)PLUS AND ( — ) MINUS ; BY TAKING CONTINUAL ARITH-METICAL MEANS BETWEEN THE SUCCESSIVE SUMS, ANDTHEIR MEANS.

ARTICLE 1.

The remarkable difference between the facility whichmathematicians have found, in their endeavours to determinethe values of infinite series, whose terms are alternately affirm-ative and negative, and the difficulty of doing the same thing■with respect to those series whose terms are all affirmative,is one of those striking circumstances in science which weCan hardly persuade ourselves is true, even after we have seenmany proofs of it; and which serve to put us ever afteron our guard not to trust to our first notions, or con-jectures, on these subjects, till we have brought them to thetest of demonstration. For, at first sight it is very naturalto imagine, that those infinite series whose terms are all affirm-ative, or added to the first term, must be much simpler intheir nature, and much easier to be summed, than those whoseterms are alternately affirmative and negative; which, how-ever, we find, on examination, to be directly the reverse;the methods of finding the sums of the latter series being nu-merous and easy, and also very general, whereas those thathave been hitherto discovered for the summation of the formerseries, are few and difficult, and confined to series whoseterms are generated from each other according to some par-ticular laws, instead of extending, as the other methods do,