and their values, except when m is equal to 1, are 1-m 1-m 21--a1 (logx)1TM-(loga)1-m (log logx)1-" — (log loga) 1-m 1-m 1-m in which x = ∞ . All these expressions are infinite if m be less than 1, and finite if m be greater than 1. If m = 1 the integrals assume the forms log x-loga, log log x-log loga, log log log x-log log log a &c. and still become infinite with x. Thus the series are convergent if m > 1 and divergent if m≥1. Perhaps there is no other mode so satisfactory for establishing the convergency or divergency of a series as the direct application of Cauchy's proposition, when the integration which it involves is possible. But, as this is not always the case, the construction of a system of derived rules not involving a process of integration becomes important. To this object we now proceed. First derived Criterion. 8. PROP. IV. The series u+u,+u2+... ad inf., all whose terms are supposed positive, is convergent or divergent according as the ratio + tends, when x is indefinitely increased, Их to a limiting value less or greater than unity. Let h be that limiting value; and first let h be less than 1, and let / be some positive quantity so small that h + k shall also be less than 1. Then as Ux+1 tends to the limit h, it is Их Иx+1 possible to give to x some value n so large, yet finite, that for that value and for all superior values of x the ratio shall lie within the limits h+k and h-k. Hence if, beginning with the particular value of x in question, we construct the B. F. D. Их 9 each term after the first in the second series will be intermediate in value between the corresponding terms in the first and third series, and therefore the second series will be intermediate in value between which are the finite values of the first and third series. And therefore the given series is convergent. On the other hand, if h be greater than unity, then, giving to k some small positive value such that h-k shall also exceed unity, it will be possible to give to a some value n so large, yet finite, that for that and all superior values of x, shall lie between h+k and h-k. Here then still each Их term after the first in the second series will be intermediate between the corresponding terms of the first and third series. But h+k and h-k being both greater than unity, both the latter series are divergent (Ex. 1). Hence the second or given series is divergent also. t2 Ex. 3. The series 1+t+ + 1.2 1.2.3 + &c., derived from the expansion of e, is convergent for all values of t. and this tends to 0 as x tends to infinity. is convergent or divergent according as t is less or greater than unity. and this tends, ≈ being indefinitely increased, to the limit t. Accordingly therefore as t is less or greater than unity, the series is convergent or divergent. If t=1 the rule fails. Nor would it be easy to apply directly Cauchy's test to this case, because of the indefinite number of factors involved in the expression of the general term of the series. We proceed, therefore, to establish the supplemental criteria referred to in Art. 5. Supplemental Criteria. 9. Let the series under consideration be ... ad inf.......... (10), Ua+Wa+2+Wa+z+Wa+s+... ad inf. the general term u, being supposed positive and diminishing in value from xa to x=infinity. The above form is adopted as before to represent the terminal, and by hypothesis positive, portion of series whose terms do not necessarily begin with being positive; since it is upon the character of the terminal portion that the convergency or divergency of the series depends. It is evident that the series (10) will be convergent if its terms become ultimately less than the corresponding terms of a known convergent series, and that it will be divergent if its terms become ultimately greater than the corresponding terms of a known divergent series. Compare then the above series whose general term is Их the first series in (8), Ex. 2, whose general term is 1 m. xm with Then m being greater than unity, and x being indefinitely increased. x being indefinitely increased, and m being equal to or less than 1. But this gives It appears therefore that the series is convergent or divergent according as, x being indefinitely increased, the function approaches a limit greater or less than unity. But the limit being unity, and the above test failing, let the comparison be made with the second of the series in (8). For convergency, we then have as the limiting equation, m being greater than unity. Hence we find, by proceeding as before, And deducing in like manner the condition of divergency, we conclude that the series is convergent or divergent according as, Should the limit be unity, we must have recourse to the third series of (8), the resulting test being that the proposed series is convergent or divergent according as, a being indefinitely The forms of the functions involved in the succeeding tests, ad inf., are now obvious. Practically, we are directed to construct the successive functions, |