TRACT S.

INFINITE SERIES.

201

obtain sets of terms approaching still nearer to the value,thus:

columns, is still more visible, and the deviations restrictedwithin narrower limits, the terms in the oblique ascendinglines still on one side of the value, and gradually increasing,while the columns and the oblique descending lines, for themost part, have their terms alternately too great and toolittle, as is evident from their alternately becoming greaterand less than each other: and from an inspection of thewhole, it is easy to pronounce that the first three figures ofthe number sought, will be 298. Taking therefore the lastfour terms of the first descending line, and proceeding asbefore, we have

And, finally, taking the lowest ascending line, because it hasmost the appearance of being alternately too great and toolittle, proceed with it as before, thus :

where the numbers in the lines and columns gradually ap-proach nearer together, till the last mean is true to the nearestunit in the last figure, giving us •298174 for the value of theproposed hypergeometrieal series 1 — 3 + 12—60 + 360 —2520 + 20160 — &c.

And in like manner are we to proceed with any other serieswhose terms have alternate signs.

Royal Military Academy,

Woolwich, May, 1780,

25

296875

296875

305664

277832

343201

361328

194336

492066

34375

25

296875

301271

291748

310516

29S073

296509

301132

297791

298821

298306

Here the approach to an equality, among all the lines and

296S75

299073

297791

298306

297974

298432

298048

298203

298240

298222