a 7. Prop. IV. To expand Eu, and "u, in a series proceeding by successive differences of some factor of Uz It will be seen that the formula of (11) page 74 and Ex. 11 page 85, accomplish this object. We shall only treat here of the very important case when Uz = a'd (x) and more especially regard the form which the result takes when a = -1, i.e. when the series is $ (0) – (1) +$ (2) - &c. We have in general, £a** (a) = (E – 1)*a** (x) = a* (a E-1) (2) (note, page 73) ao aA $ A? 1 + - &c. c). 2 2 4 This enables us to transform many infinite series into others of a more convergent character; for $ (0) -$(1) + &c. ad inf. + &c. (0) (21, 1 + E 4 which is very rapidly convergent if the other is but slowly so. 1 1 1 Ex. Transform the series &c. into a 12 13 14 more convergent form. Here $ (0) = (0 +12)-1), .. we have by (21) 1 1 1 (1 1 2 + &c. + 12 13 2 12 2.12.13 4.12.13.14 2.3 + + &c 8.12.13.14.15 which converges rapidly. + + + &c.}, 8. It is very often advisable to find the sum of the first few terms of a series by ordinary addition and subtraction, and then to apply our formulæ to the remaining terms, as in this way the convergence of the resulting series is usually greater. Thus, if we had applied the formula just obtained to the series 1 1 1 + &c., we should have obtained 1 2.3 + &c. &c.}; a much more slowly converging series. This remark is of great importance with reference to all the formulæ of this Chapter. We shall see that the Maclaurin Sum-formula of Art. (2) usually gives rise to series that first converge and then diverge, but that by keeping only the convergent part we obtain an approximate value of the function on the left hand side of the identity; and also that the closeness of the approximation depends on the smallness of the first of the terms in the rejected portion. From this it follows that by applying the formula in the manner just indicated we can greatly increase the closeness of the approximation. An example will make it clearer. Taking this between limits 2 and 1, we obtain 1 1+ 1 1 9 + &c. =1+3+B, -B,+ B, &c. Now, remembering that we must only keep the convergent part of the series, we find that we must stop at B, since after that the numbers begin to increase. This gives us + Σ x = 5 2 BY + 3 + &c. 1.65714, the true value being or 1.64493. 6 Now let us find the sum thus 1 1 1 1 1 x = 1 ++ &c. ad inf. =1+ + + 4 9 4 9 16 B. 5° On examination it will be found that we may in this case keep the terms at least as far as B..*, while the convergence is so rapid at first that by only retaining as far as B, we obtain 1.64494. The general advantage of using the formula may be gathered from this example. To obtain an equally close approximation by actual summation, some hundred thousand terms would have to be taken. 9. We can also expand Ea*d (w) in a series proceeding by successive differential coefficients of $ (Q). For Sa+ (z) =(E-1)*a*6(z) = a* (E-1)^4 (w)...... (23). But by Herschel's Theorem q (e) = 4 (E) eore, .: 4 (E) = 74 (e") = 4 (E') eo.D as operating factors, where E' affects () only, 02 :: Ea*b (X) = a* (a. E' – 1)-3 (1 +0.D+ Do + &c. $ (2) 1.2 + & 1.2 dic doc sa E 1 where A, E aA a In the case of a=-1 an expression for A, in terms of Bernoulli's numbers can be obtained. For E(-1)** («) = (– 1)*(- € – 1)* (), putting a=-1 =(-1)*-1 (e” + 1)*(x). { {1+4, 4, $(*) + &c. I. (24) + , a in (23), B29 * In reality we may keep all terms up to a quantity whose first significant figure is in the fourteenth decimal place. 531) 2 1 B, B, [ 2 10. Expansion in inverse factorials. The most general method of obtaining such expansions is by expressing the given function $(x) in the form e-*f(t) dt. If we then write 0 dz. (*=1–2, we get $ (2) = (1 – 3)=uf{log (1 - 2)} s{log (1-2)} e e 1 must now be expanded in some way in powers 11 of , and each term must be integrated separately by means of the formula im 2 (3C + 1)...(c + m) By performing & on this we can expand in a similar way the more complicated form f (t) dt. The most in 1 teresting cases are those in which $ (x) = log æ or = (see page 115). The method is obviously very limited in its application. A paper on it by Schlömilch will be found in Zeitschrift für * Compare (7), page 108. Ex. 12, page 85, is closely connected with the problem of this article. e 0 . & Math. und Physik, iv. 390, and a review of this in Tortolini (Annali, 1859, 367) has sufficiently copious references to enable any one who desires it to follow out the subject. Stirling's formula—the earliest of the kind—is given in Ex. 11, page 30. The very close connection that Factorials in general have with the Finite Calculus renders it worth while to give special attention to them, and to investigate in detail the laws of their transformations. For this purpose the student may consult a paper by Weierstrass (Crelle, LI. 1). Oettinger has also written on the subject (Crelle, xxxIII. and XXXVIII.), and Schläfli (Crelle, XLIII. and LxvII.). Ohm has an investigation into the connection between them and the Gamma-function (Crelle, xxxvI.), with a continuation on Factorials in general (Crelle, xxxix.). The papers on the subject of the Euler-Maclaurin Sum-formula are very Characteristic examples have been selected from them where it was possible, and placed, with references, in the accompanying Exercises. By far the most important application of the principle of approximation is to the evaluation of Tx, or rather of log fx and its differential coefficients when x is very large. Raabe has two papers on this (Crelle, xxv. 146 and XXVIII. 10). See also Bauer (Crelle, LVII. 256) and Guderman (Crelle, xxix. 209). Reference will be made to these papers when we consider Exact Theorems. See also a paper by Jeffery (Quarterly Journal, vi. 82) on the Derivatives of the Gamma-function. The constant C of Ex. 3 is of great importance in this theory. For its value, which has been calculated to a great number of decimal places, see Crelle, Lx. 375. Closely connected with the subject of differential coefficients of log Tac is . 1 that of the summation of harmonic series ( On this see {a+ (n - 1)d}" papers by Knar (Grunert, XLI. and XLIII.). numerous. - 1)dyr). EXERCISES. 1. Find an expression for 1 1 1 + 12 52+g2+ &c., to n terms, and obtain an approximate value for the sum ad infinitum. 2. Find an approximate expression for and also the value of 1 1 25 +35+ &c., ad inf., to 10 places of decimals. 1 |