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and their values, except when m is equal to 1, are
in which x=0. All these expressions are infinite if m be
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, +... ad inf., all whose terms are supposed positive, is convergent or divergent according as the ratio 4x+1 tends, when x is indefinitely increased,
Let h be that limiting value; and first let h be less than 1, and let k be some positive quantity so small that h + k shall also be less than 1. Then as
Ug+1 tends to the limit h, it is possible to give to x some value n so large, yet finite, that for that value and for all superior values of æ the ratio 4x+1 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.
Win + (h + k) Um + (h + k) un + &c.
(9), Un + (k – k) Un +(h – k) Un + &c. 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
1- (h + k) 1-(h – k)' 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 w some value n so large, yet finite, that for that and all superior values of x, Ug+1 shall lie between h + k and h-k. Here then still each Ux 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.
+ t Ex. 3. The series 1+++ + + &c., derived
1.2 1.2.3 from the expansion of e', is convergent for all values of t,
and this tends to () as a tends to infinity.
Ex. 4. The series
a (a +1) ++
a (a + 1) (a + 2) tt
t + &c. b (b + 1) 6 (b + 1) (6 + 2) is convergent or divergent according as t is less or greater than unity.
a (a+1) (a + 2) ... (a + x - 1) Here
t" 6 (6+1) (6 + 2) ... (6 + 3 — 1)
Ux and this tends, ą 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.
b + x
Un + Watt + Wate+ Ua+3+ ad inf. ......... (10),
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 Ux with
1 the first series in (8), Ex. 2, whose general term is Then a condition of convergency is
m being greater than unity, and x being indefinitely increased.
a 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, æ being indefinitely increased, the function Tog æ 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,
2 (log x)" m being greater than unity. Hence we find, by proceeding as before,
1 log хих
log log 2 And deducing in like manner the condition of divergency, we conclude that the series is convergent or divergent according as,
log x being indefinitely increased, the function tends to
log log a a limit greater or less than unity.
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,
æ being indefinitely 1 log
a log xux tends to a limit greater increased, the function
log log log a or less than unity.
The forms of the functions involved in the succeeding tests, ad inf., are now obvious. Practically, we are directed to construct the successive functions, 1 1
1 1 7 1
&c. .... (A),