tract 8.

INFINITE SERIES.

181

it follows that s lies between a and a — b, and that the arith-metical mean a — is something above the value of s, butnearer to that value than a is. And thus, the same reason-ing holding in every following pair of successive sums, thearithmetical means between them will form another series ofterms, -which are, like those sums, alternately less and greaterthan the value of the proposed series, but approximatingnearer to that value than the several successive sums do, asevery term of those means is nearer to the value s, than thecorresponding preceding term in the sums is. And, like asthe successive sums form a progression approaching alwaysnearer and nearer to the value of the series; so, in like man-ner, their arithmetical means form another progression, com-ing nearer and nearer to the same value, and each term of theprogression of means nearer than each term of the successivesums. Hence then we have the two following series, namely,of successive sums and their arithmetical means, in which eachstep approaches nearer to the value of s than the former, thelatter progression being however nearer than the former, andthe terms or steps of each alternately below and above thevalue s of the series a — b -)- c — cl -f- &c.

Successive sums.

"3 0c- a

“a a — bc- a — b + c"3 a — b + c — de~ a — b + c — d +e&c.

Arithmetical means,

“a 4 a

c a — 4 b

-3 a — b + \c

c- a — b + c —

“3 a — b + c — d + 4r

it- a— b + c— d+ t —\f

&c.

where the mark -a, placed before any step, signifies that itis too little, or below the value s of the converging seriesa b c — d &c; and the mark r- signifies the con-trary, or too grcat> ^nd h ence ’_ a> 0 r half the first term

of such a converging series, is less than s the value of the

series.