168 .

THE NATURE AND VALUE

TRACT 7.

whose terms continually increase. So that a series whoseterms neither increase nor decrease, but are all equal, as theyneither converge nor diverge, may be called a neutral series,as a — a a — a - 1- &c. Now converging series, being sup-posed infinitely continued, may have their terms decreasingto 0 as a limit, as the series 1 — •£ + 4 — + + & c > or on ^ydecreasing to some finite magnitude as a limit, as the seriesr — i + T — 4 + &c, which tends continually to 1 as alimit. So, in like manner, diverging series may have theirterms tending to a limit, that is either finite or infinitelygreat: thus the terms 1 — 2 + 3- 4 + &c, diverge to in-finity ; but the diverging terms f — •§• + J- — 4 + &c, onlyto the finite magnitude 1. Hence then, as the ultimate termsof series which do not converge to 0, by supposing themcontinued in infinitum , may be either finite or infinite, therewill be two kinds of such series, each of which will be furtherdivided into two species, according as the terms shall eitherbe all affected with the same sign, or have alternately thesigns + and —. We shall, therefore, have altogether fourspecies of series which do not converge to 0, an example of

each of which may be as here follows:

r

1 + 1 + 1 + 1+ 1+ 1 + &c.

++4+4+4+ 4 + 4 + &C.

1 - I + 1 - 1 + 1 - 1 + &c.

4 - 4 + I - 4 + i - 4 +

1 + 2 + 3+ 4+ 5 + 6 + &c.

1 + 2 + 4 + 8 + 16 + 32 + tkc.

1 — 2 + 3— 4+ 5 - 6 + &c.

1 _ 2 + 4 - 8 + 16 - 32 + &c.

3. Now concerning the sums of these species of series,there have been great dissensions among mathematicians;some affirming that they can be expressed by a certain sum,while others deny it. In the first place, however, it is evi-dent that the sums of such series as come under the first ofthese species, will be really infinitely great, since by actually