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If therefore, the expression for at a be
a', C., + æ, C., + ...

that for the sum of the series is discontinuous, and represented by

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I shall take the two instances given in my former paper, which will of course be the most satisfactory, as the difficulty was prior to the explanation. The first was the series

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a particular solution of pa = q (a”) being pa: = a : = va: ;

if a lie between – 1 and + 1, is r.

To explain the cases where aa = a, return to the expression for the series, which then becomes

aa $1 — ("ya)*},
giving in this case for a = 0, the value 0,
and for a = 1 the value 1.


The result may be easily verified. The given series may be thrown

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The second example in my former paper was the series
(1 or a” as
+ ...

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The equation of the series is
q, w = (TTE) (TTa, + p (aa),

a particular solution of which is aa = (a-I) (ITI)”

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when a < 1 = ( This result may also be verified; for the original series developed

term by term in powers of ; and }. and corresponding series of

powers of . collected gives

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and resolved into series of positive powers of a and ar, with a similar subsequent process, it yields

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Let us now apply this method to examine some of the more common series of analysis: let us take

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Multiply every term by a , and it will then appear to be

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Like all other results of strict methods of passing from the sum of n terms to the sum of an infinite series, this expression is infinite when the series is infinite. But my object here is to remark, that owing to a "a = a having only the root a = 0, there can be no discontinuity among the values which correspond to arithmetical values of the above series.

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we find the value of the preceding to be as follows,

I A particular solution of par = ***) is va: = T (a + 1). No more simple value of air can be found than the finite integral of Au o (to * Igo. – T 1...g. 3..." If u a could be expressed, the value of the preceding series would be a (<) pu a' – T. 2. 3.5- (1 ... 2 . 3...ar).

What we have here to observe is, that in consequence of air = a + 1, a” a increases without limit for all values of ar, and there is no dis


I shall only further remark, that the preceding results confirm, so far as they go, an opinion which I have long entertained, namely, that series which may be divergent, or which may be brought as near to divergency as we please, such as that for e", require much less circumspection than those which can never be made to diverge. In the first, generally speaking, the arithmetical value (between the limits of convergency) is the analytical value throughout; in the second, there is frequently discontinuity in the arithmetical values, and the general equivalent of analysis is not easily expressed. I will not however enlarge upon so general a topic, but beg to remain,

Dear Sir,
Yours very truly,

5 Upper Gower Street,

April 30, 1836.

. P. S. Some time ago, I communicated to the Society what I consider a failure in the proof of the celebrated theorem of M. Abel, on the expressibility of the roots of equations which are values of a periodic function. As I have since printed my objection in the Calculus of Functions alluded to in the preceding paper (Ş. 90, 302, 303,) I take this opportunity of referring to the subject.

Vol. VI. PART I. B B

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