17. Find the condition that the product u, u, Ug ...... should be finite. Ex. 21.31.41 18. If the series u, + u, tu, + .... has all its terms of the same sign and converges, shew that the product (1+x)(1 + )...... is finite. Shew that this is also the case when the terms have not all the same sign provided the series and that formed by squaring each term both converge. [Arndt, Grunert, XXI. 78]. CHAPTER VIII. EXACT THEOREMS. case. 1. In the preceding chapters and more especially in Chapter 11. we have obtained theorems by expanding functions of A, E and D by well-known methods such as the Binomial and Exponential Theorem, the validity of which in the case of algebraical quantities has been demonstrated elsewhere. But this proceeding is open to two objections. In the first place the series is only equivalent to the unexpanded function when it is taken in its entirety, and that is only possible when the series is convergent; so that there can in this case alone be any arithmetical equality between the two sides of the identity given by the theorem. It is true that the laws of convergency for such series when containing algebraical quantities have been investigated, but it is manifestly impossible to assume that the results will hold when the symbols contained therein represent operations, as in the present And secondly, we shall very often need to use the method of Finite Differences for the purpose of shortening numerical calculation, and here the mere knowledge that the series obtained are convergent will not suffice; we must also know the degree of approximation. To render our results trustworthy and useful we must find the limits of the error produced by taking a given number of the terms of the expansion instead of calculating the exact value of the function that gave rise thereto. This we shall do precisely as it is done in Differential Calculus. We shall find the remainder after n terms have been taken, and then seek for limits between which that remainder must lie. We shall consider two cases only—that of the series on page 13 (usually called the Generalized form of Taylor's Theorem) and that on page 90. The first will serve for a type of most of the theorems of Chapter II. and deserves notice on account of the B. F. D. 10 relation in which it stands to the fundamental theorem of the Differential Calculus; the close analogy between them will be rendered still more striking by the result of the investigation into the value of the remainder. But it is in the second of the two theorems chosen that we see best the importance of such investigations as these. Constantly used to obtain numerical approximations, and generally leading to divergent series, its results would be wholly valueless were it not for the information that the known form of the remainder gives us of the size of the error caused by taking a portion of the series for the whole. Remainder in the Generalized form of Taylor's Theorem. 2. Let v be a function defined by the identity (oc – a) v. = Uz — Wa. (1). By repeated use of the formula Aw-V=Wx+2 Avr+vxAWx.... (2) we obtain (α - α + 1) Δυ, +υ, = Auz, (a – a + n) A": + nano)0 = A"Uz. Substituting successively for Vx, Avz, A0. ... we obtain after slight re-arrangement (a – ) (a – X – 1) A’uz+ &c. in ..(3), where R = (a — «) (a — *— 1) ... (a — x − n) Allox .....(4), Un + in ........ a 3. This remainder can be put into many different forms closely analogous, as has been said, to those in the ordinary form of Taylor's Theorem. For instance, if u, =f (a) we have =A" f'(x + (a — «) 0}, where 0 is some proper fraction. If we write x + h for a, this last may be written Auf (x + ho) where Ax is now supposed to be 1-instead of unity, and R, appears under the form (*+1) (Δα)" .(5), from which we can at once deduce Cauchy's form of the remainder in Taylor's Theorem, i.e. 2*41 (1 - 0)" f**? (oC+0h), In after the easy generalization exemplified at the bottom of n page 11. 4. Another method of obtaining the remainder is so strikingly analogous to one well known in the Infinitesimal Calculus that we shall give it here. (Compare Todhunter's Diff. Cal. 5th Ed. p. 83.) Let — ) φ(α) - φ (α) - (α - α) Δφ (α) - . 12 n ) Δ*Φ () Then, since from (2) Δφ () - r (= (2—3— 1) (-1) Δ'φ(α), r-1 we obtain n ΔF (α) -A**l* () ............ (6). Now if > - « be an integer F(x) – F() = AF (0) + AF (x + 1) +...+AF(2– 1)......(7), and hence is equal to the product of (2 — 2) and some quantity intermediate between the greatest and least of these quantities, and as AF(x) is supposed to change continuously through the space under consideration, it will at some point between x and 2 (we might say between x and 2-1), take the value in question, and we may thus write (7), F(x) – F () = (2 — 2) AF{z +0 (ic — 2)}. But F (2)= 0, ::(6) becomes F(x) =-(2 — 2) AF {z + 0 (oc — 2)} _(3 — ) {O (3 — «) – 1)" 4n+18+0 (2 — 2)}, or, if % - x=h, A*+1 (c+h-oh) .(8). өп n 5. A more useful form of the result would be derived at once by summing both sides of (6), remembering that F(z) is zero. Since (2 - x - 1)() is positive for all values of w less than ź, we see that F (9) lies between the products of the sum of the coefficients of the form (z – X – 1)(") by the greatest and ir least values of Ant]$ (ə). But the sum in question is so that the form thus obtained is very convenient. (2 — «)(n+1 n +1 |