EXERCISES. 1. Find, by an application of the fundamental proposition, Art. 4, two limits of the value of the series In particular shew that if a = 1, the numerical value of the series will lie between the limits π and 7. 4 +&c. is convergent if x is equal to or less than unity, divergent if x is greater than unity. 3. Prove, from the fundamental proposition, Art. 4, that the two series $ (1) + $ (2) + $ (3) + &c. ad inf. are convergent or divergent together. 4. The hypergeometrical series ab a (a+1)b(b+1) 1+ x+ cd c (c + 1) d (d+1) is convergent if x < 1, divergent if x > 1. m being positive, x2+ &c. If x=1 it is convergent if c+d-a-b>1, divergent if c+d-a-b1. CHAPTER VI. THE APPROXIMATE SUMMATION OF SERIES. 1. IT has been seen that the finite summation of series depends upon our ability to express in finite algebraical terms, the result of the operation Σ performed upon the general_term of the series. When such finite expression is beyond our powers, theorems of approximation must be employed. And the constitution of the symbol Σ as expressed by the equation Σ = (e2 – 1) ̃1 ... (1), d renders the deduction and the application of such theorems easy. Speaking generally these theorems are dependent upon the d development of the symbol Σ in ascending powers of da or, under particular circumstances, in ascending powers of A. Hence two classes of theorems arise, viz. 1st, those which express the sum of a series whose general term is given, by a new and rapidly convergent series proceeding according to the differential coefficients of the general term, or to the differential coefficients of some important factor of the general term; 2ndly, those which differ from the above only in that differences take the place of differential coefficients. The former class of theorems is the more important, but examples of both and illustrations of their use will be given. 2. PROP. To develope Eu, in a series proceeding by the differential coefficients of uz· d dx d Since Σu2 = (eTM – 1) ̃1u, we must expand (1) in ascending powers of dx' d and the form of the expansion will be determined by that of the function (e-1). For simplicity we will first deduce a few terms of the expansion and afterwards determine its general law. Now For most applications this would suffice, but we shall proceed to determine the law of the series. The development of the function (e' — 1)1 cannot be directly obtained by Maclaurin's theorem, since, as appears from (1), it contains a negative index; but it may be obtained by by Maclaurin's theorem and dividing the t expanding €-1 result by t. t Referring to (1) we see that the development of will €* 1 havet for its second term. It will now be shewn that this is the only term of the expansion which involves an odd power of t. Let R being the sum of all the other terms of the expansion. Then B. F. D. 6 and as this does not change on changing t into t, the terms represented by R contain only even powers of t. Now if for the moment we represent e 1 by 0, we have + &c. =1 But by the secondary form of Maclaurin's theorem, Chap. II. since ▲"0" vanishes when n is greater than m. It is to be noted that when m = O we have 0 = 1 and m m] = 1. The expression (4) determines in succession all the coefficients of the development of t (e-1) in ascending powers of t. It must therefore, and it will, vanish when m receives any odd value greater than 1. From these results we may conclude that the development of (e* — 1)1 will assume the form where in general A, is expressed by (4). It is however customary to express this development in the somewhat more arbitrary form B2r-1 The quantities B1, B, &c. are called Bernoulli's numbers, and their general expression will evidently be B3»-1 = (− 1)TM+1 { — † A0o + } A30o ... + ...(7). 2r+1) Or, actually calculating a few of the coefficients by means of the table of the differences of 0 given in Chap. II., 3. Before proceeding to apply the above theorem a few observations are necessary. Attention has been directed (Differential Equations, p. 376) to the interrogative character of inverse forms such as The object of a theorem of transformation like the above is, strictly speaking, to determine a function of a such that if we perform upon it the corresponding direct operation (in the d above case this is -1) the result will be ux. To the inquiry what that function is, a legitimate transformation will necessarily give a correct but not necessarily the most general answer. Thus C in the second member of (8) is, from the mode of its introduction, the constant of ordinary integration; but for the most general expression of Eu, Cought to be a |