OF AN INFINITE ANGLE. 261 



to this also Mr. De Morgan has given assent where he assumes that 1 — 1+1 — ... is a form of 



I — a;' + a:* — ay' + 0;^'^ — I have brought forward these testimonies, because it is not very 



unusual to cast a mantle of mystery over this subject, by introducing zeros into the expansion of 



. But such a device, however much it may serve to satisfy the eye, cannot satisfy the 



head : for — gives 1 - x + x^ - x' + ..., there being no terras between x and a'', a-' and .v"^, 



1 T (37 + J. 



&c., in this, which is the general form of the series ; and consequently it is not allowable to write 



= 1-1 + 0+1-1+0+..., if it be intended to insinuate thereby that the zeros make 



1+1+1 ^ 



any difference in the sum of the infinite series: and if they make no difference, why introduce them.-' 



11. On principles therefore which are allowed, and used by the writers quoted, it is established 

 that 1 -1 + 1 — ... has no definite equivalent, in the sense in which this word is generally under- 

 stood. I think also it is proved, that is in no proper sense the equivalent of 1 — ,v + x" — ..., 



1 + iT7 



except when this series is convergent. For that the two expressions may be equivalent to each other, 

 it is essential that each should exhibit the same degree of indeterrainateness of value in particular 

 cases, and the same kind of discontinuity : but, as we bave seen, there is no such agreement : on the 



contrary, while it is admitted that, as x converges towards 1, approaches towards i as its 



1 + X ** 



unique limit, it is here shewn that the other member of the assumed equivalence approaches towards 



an indeterminate form of an ambiguous character, and absolutely refuses to approach in any case to 



1 - 1 + 1 - ... as a limit of continuity. 



12. It is not the purpose of this paper to treat of Diverging Series in general, but only of the 

 recurring form 1 — 1 + 1 — ... , and of this only because it has been connected with the values 

 of Sin OS and Cos os , yet as the method above employed is applicable to the general form 

 0a; = flia." + Ooa,-^ + ... + a,,x'' + ... I may state that the same mode of reasoning when applied to 

 this, shews that (bx does not embody the algebraical properties of the series, unless the value of x, 

 and the form of the coefficients, be such as to make a^x" tend to zero as its limit when ti and v 

 approach co . Series which satisfy this test 1 call convergent series, whether the arithmetical sum 

 thereof be finite or infinite : and all such series are distinguished by this property, that their invelopcs 

 may be safely used as equivalent to them in every sense both algebraical and arithmetical. 



13. From this it is evident, that the operation of integration performed upon a series will often 

 (not always) have the effect of removing its discontinuity, and establishing a real equivalence though 

 none existed before. And so the operation of differentiation will not unfrequently have the effect of 

 introducing discontinuity, and destroying equivalence. 



Hence we see why we may put 1 for x in log^ (l + x) = x -{ • — ..., though we may not 



write 1 for x in =1 — a; + x'' — ... from which it was derived by integration. 



14. But, in pursuance of the object of this memoir, it is time now to turn to the series 



I - Cos0 + Cos20 - Cos 30 + ... whicii has been assumed to be a form which can approach 



I - 1 + 1 - as a limit by diminishing 9 towards zero. Now assume y to be any arbitrary 



y ... 



Jiniie angle, and put = ± - which will be indefinitely small for the terms whore w is infinite. 



Hence in such terras Coan6= Cos ± y = Cosy= a. finite quantity, not equal to unity, because y 



