Ex. 2. Let it be required to develope Euv, in a series proceeding according to Eva, vx› &c. In applying this theorem, we are not permitted to introduce unconnected arbitrary constants into its successive terms. If we perform on both sides the operation A, we shall find that the equation will be identically satisfied provided AΣ"uz in any term is equal to 1u in the preceding term, and this imposes the condition that the constants in Σu be retained without change in "u. And as, if this be done, the equation will be satisfied, it follows that however many those constants may be, they will effectively be reduced to one. Hence then we may infer that if we express the theorem in the form Σuxvx = C + ux-2 Σvx − Aux, Σ2vx+ A2ux-2 Σ3vx.............. (1), we shall be permitted to neglect the constants of integration, provided that we always deduce "v, by direct integration from the value of "v, in the preceding term. If u„ be rational and integral, the series will be finite, and the constant C will be the one which is due to the last integration effected. We have seen that C is a constant as far as ▲ is concerned, i. e. that AC-0. It is therefore a periodical constant going through all its values during the time that x takes to increase by unity. The necessity of a periodical constant C to complete the value of Eu may also be established, and its analytical expression determined, by transforming the problem of summation into that of the solution of a differential equation. d Let Zuy, then y is solely conditioned by the equation Ayu, or, putting ed-1 for A, by the linear differential equation d Now, by the theory of linear differential equations, the complete value of y will be obtained by adding to any particular value v, the complete value of what y would be, were u equal to 0. Hence C, C, &c. being arbitrary constants, and m,, m,, &c. the different roots of the equation €TM-1=0. Now all these roots are included in the form m = ± 2iπ √− 1, i being 0 or a positive integer. When i=0 we have m = 0, and the corresponding term in (2) reduces to a constant. But when i is a positive integer, we have in the second member of (2) a pair of terms of the form which, on making C+C' = A,, (C−C')√−1 = B1, is reducible to 4, cos 2iπ + B; sin 2iT. Hence, giving to i all possible integral values, ΣUx=Vx+C+A ̧ cos 2πx + А, cos 4πж+  ̧ соs бπx+ &c. 2 3 +В, sin 2πx + В, sin 4πx + B, sin 6πx + &c. (3). 1 2 ...... The portion of the right-hand member of this equation which follows vx is the general analytical expression of a periodical constant as above defined, viz. as ever resuming the same value for values of x, whether integral or fractional, which differ by unity. It must be observed that when we have to do, as indeed usually happens, with only a particular set of values of x progressing by unity, and not with all possible sets, the periodical constant merges into an ordinary, i. e. into an absolute constant. Thus, if x be exclusively integral, (3) becomes ΣUx=Vx+ C + A ̧ + A‚† à ̧+&c. =Vx+c, c being an absolute constant. 1 2 It is usual to express periodical constants of equations of differences in the form (cos 2πα, sin 2πα). But this notation is not only inaccurate, but very likely to mislead. It seems better either to employ C, leaving the interpretation to the general knowledge of the student, or to adopt the correct form C+Σ¡ (4; cos 2iπx + В¡ sin 2iπx) vi i We shall usually do the former. (4). 5. The student will doubtless already have perceived how much the branch of mathematics that forms the subject of our present consideration suffers from its not possessing a clear and independent set of technical terms. It is true that by its borrowing terms from the Infinitesimal Calculus to supply this want, we are continually reminded of the strong analogies that exist between the two, but in scientific language accuracy is of more value than suggestiveness, and the closeness of the affinity of the analogous processes is by no means such that it is profitable to denote them by the same terms. The shortcomings of the nomenclature of the subject will be felt at once if one thinks of the phrases which describe the operations analogous to the three chief operations in the Infinitesimal Calculus, i. e. Differentiation, Integration, and Integration between limits. There is no reason why the present state of confusion should be permanent, so that we shall in future (in the notes at least) denote these by the unambiguous phrases, performing ▲, taking the Difference-Integral (or performing ), and summing, and shall name the two divisions of the calculus, the Difference- and the Sum-Calculus respectively, and consider them as together forming the Finite Calculus. The preceding chapters have been occupied with the Difference-Calculus exclusively-the present is the first in which we have approached problems analogous to those of the Integral Calculus; for it must be borne in mind that such problems as those on Quadratures are merely instances of use being made of the results of the Difference-Calculus, and have nothing to do with the Sum-Calculus, except perhaps in the case of the formula on page 55. Enough has been said about the analogy of the various parts of our earlier chapters with corresponding portions of the Differential Calculus, and we shall here speak only of the exact nature and relations of the Sum-Calculus. If the nth term of a series be known, and its sum be required, it is tantamount to seeking the difference-integral, and our power of finding the difference-integral is coextensive with our power of finding the sum of any number of terms. Hence the summation of all series, whose sum to n terms can be obtained, is the work of the Sum-Calculus. It is true that there are many series, that can be summed by an artifice, of which we have taken no notice, but that is not because they do not belong to our subject, but because they are too isolated to be important. But it must be remembered that the difference-integral is only obtainable when we can find the sum of any number of consecutive terms we may wish. But there are many cases in which we seek the sum of n terms of a series which is such that each term of the series involves n, e.g. we might desire the sum of the series 1. n+2. (n-1) + 3. (n − 2) + &c. to n terms. Now in a certain sense this is not a case of summation; we do not seek the B. F. D. 6 sum of any number of terms, but of a particular number of terms depending on the first term of the series itself. And, as might be expected, this operation has not the close connexion that we previously found with that of finding the difference-integral of any term; for though the knowledge of the latter would enable us to sum the series, yet the knowledge of the sum of the series will not enable us to find the difference-integral of any term. These must be called definite difference-integrals, and hold exactly the same position that Definite Integrals occupy in the Infinitesimal Calculus. No one would think of excluding from the domain of Integral Calculus the treatment of such functions as the definite integral x' (a-x)m dx, because the knowledge of its value does not give us any clue to that of the indefinite integral (a-x)m dx, and is obtained indirectly without its being made to depend on our first arriving at the knowledge of the latter. 0 a By similar considerations we shall arrive at a right view of the relation of infinite series to the Sum-Calculus. It is often supposed that it has nothing to do with such series-that the summation of finite series is its business, and that this is wholly distinct from the summation of infinite series. This is by no means correct. The true statement is that such series are definite difference-integrals, whose upper limit is ∞, and so far they as much belong to our subject as €-*2 dx does to the Infinitesimal Calculus. How is it then that the whole subject of series is not referred to this Calculus, but is separated into innumerable portions, and treated of in all imaginable connexions? It is that in the expression of such series as those we are speaking of, reference being only made to finite quantities, there is nothing to distinguish them from ordinary algebraical expressions, except that the symmetry is so great that only a few terms need be written down. Hence when it is summed by an artifice, and not by direct use of the laws of the Sum-Calculus, there is nothing to distinguish the process from an ordinary algebraical transformation or demonstration of the identity of two different expressions. Now in Definite Integrals that are similarly evaluated by an artifice, there is perhaps just as little claim for the evaluation to be classed as a process belonging to the Infinitesimal Calculus, but the expression of the subject of that process involving the notation and fundamental ideas of the Calculus, it is naturally classed along with processes that really belong to the Calculus. Thus the Infinitesimal Calculus has a wide field to which no recognized branch of the Finite Calculus corresponds, not because it does not exist, but because it is not reserved for treatment here. No doubt this has its disadvantages. Series would be more systematically treated, and the processes of summation more fully generalized, if they were dealt with collectively; yet on the other hand it is a great advantage in the Finite Calculus to have to do only with such processes as really depend on its laws, and not with processes that are really foreign to it, and are only connected therewith by the fact that their subject-matter in these particular instances is expressed in the form of a series, i. e. in the notation of the Calculus. It is not usual to speak of such identities as Definite Difference-Integrals, but a certain class of them are considered in this light in a paper by Libri (Crelle, XII. 240). Before leaving the subject of Definite Difference-Integrals we must mention a paper by Leslie Ellis (Liouville, IX. 422), in which he demonstrates a theorem analogous to the well-known one on the value of SSSS... f (x+y+...) dx dy đz ..., where x+y+z+...+1. The method is a very beautiful one, but we must not be supposed to endorse it as rigorous, since one part involves the evaluation of x1) cos ax. 0 The fundamental operations of the Finite Calculus are taken as ▲ with its correlative 2. In this view of the subject the sign of each term is supposed to be +, not that its algebraical value is supposed to be positive, but that its sign must be accounted for by its form. Thus if we take the series Up - Uz + U2- &c., we must call the general term (-1) u. To avoid this complication in the treatment of series whose terms are alternately positive and negative, some have wished to have a second Calculus whose fundamental operation is 1+ E, the correlative of which, -1, would of course denote the operation of summing such a series. A series of papers by Oettinger, the inventor of it, will be found in Crelle, Vols. XI.—XVI. In these he developes the new Calculus in a manner strictly analogous to that in which he subsequently treats the Difference-Calculus, connects them similarly with the Infinitesimal Calculus, demonstrates analogous formulæ, and applies them at first to simple cases and then to more complex ones, especially to those series whose terms are products of the more simple functions and those most suitable to such treatment. The work is unsymbolical, and therefore clumsy and tedious compared with more recent work, and we should not have referred to the papers here (for we consider it highly unadvisable to invent a new Calculus for a comparatively unimportant class of questions that can very easily be dealt with by our present methods) were it not that his results are very copious and detailed. The student who desires practice in the symbolical methods cannot do better than take one of these papers and employ himself in demonstrating by such methods the results there given. Should he desire however a statement of the nature and advantages of this more elaborate treatment of series, he will find it in a review by Oettinger. (Grunert, Archiv. XIII. 36.) This is not the only attempt to introduce a new Finite-Calculus. A certain class of series is treated in a paper by Werner (Grunert, Archiv. XXII. 264), by means of a calculus whose fundamental operation, A = E - v2, is almost the most general form of linear fundamental operation that can be imagined. EXERCISES. 1. Sum to n terms the following series: 1.3.5.7+3.5.7.9+... |