34 +&c. &c.} 24 203 1 1 1+ + (8). Equating the coefficients of the same powers of 0 in (5) and (8), we obtain 2 1 1 2n Bon-1 ...)= 2n-1 22n, ... 21 2n 1 1 1+ + + :(9)* (27) 22n 32n From this we see that the values of Ban-, increase with B. very great rapidity, but those of in- ultimately approach to 2n equality with those of a geometrical series whose common 1 ratio is 47" * A variation of (9), due I believe to Raabe (Diff. und Int. Rechnung, 1. 412), depends on the following ingenious transformation: 1 1 1 S=1+ + 22n 32n + 42n +&c.; 1 and all the terms of the form are removed. Proceeding as before (2p)2n 1 1 52n where 2, 3, 5 ... are the prime numbers taken in order. This formula would be of great use if we wished to obtain approximate values of Bri corresponding to large values of n, as it is well adapted for logarithmic computation. 4. If m be a positive integer and P be positive 5. Euler was the first to call attention to a set of numbers closely analogous to those of Bernoulli. They appear in the coefficients of the powers of a when sec w is expanded. Thus E EM (11). 2 4 0 A2n. do .(14), te formulæ analogous to (9) and (10), from which (12) may be deduced. * Due to Plana (Mem. de l'Acad. de Turin, 1820). 6. Owing to the importance of Bernoulli’s and Euler's numbers a great many different formulæ have been investigated to facilitate their calculation. Most of these require them to be calculated successively from B, and E, onwards, and of these the most common for Bernoulli's numbers is (2). Others of a like kind may easily be obtained from the various expansions which involve them. Thus from (5), multiplying both sides by sin 0, A3 B, B. 3 o 2 4 and equating coefficients of Aan we obtain (-1)" 22n-2 (-1)"} BO Bm-s+ &c. 2n 2n 32n - 2 2n +1 22n 2n-1 .(15). The simplest formulæ of this nature both for Bernoulli's and Euler's numbers are obtained at once from the original assumptions t t 1 e-1 1 + Σ cos t by this method. 1--:(-1)" Bn2_1421 and 2n 2n 7. But direct expressions for the values of the numbers may be found. Thus t log et log E 20++ (by Herschel's theorem) e-1 et - 1 E-1 Hence, equating coefficients, we find (-1)"+1 B2n-1 log (1 +A) 021 ; 2n A and in like manner we obtain 0 = {1 A - &c. &c: 02n+1 (n > 0) ............ (17). 8. These formulæ are capable of almost endless trans Δ*Ο* formation. Thus, since An-10--1 = A"0*-1 (Ex. 8, page 28), we can write (16) thus n 49 &c.) 02nt:1 247 32 Bra=(-1)** (-- &c.) on? AR + ..(18), since the other term is log (1+A) 02* = D021 = 0. 9. A more general transformation by aid of the formula f(A)0" = Ef'(A) 01-1 is as follows: {log (1+yE)} Of(0) = yf(1) –... 28 (2) + &c. -1 (20), if f(0) = 0. In (19) write E' for x and operate with each side on f(0). 1+yĘS ( 1+yĒf (0) Then A {log (1 + AE')} O" f (0') : 1 +AE0"-1 f(0) — {log (1 + AE')} 07-10'A' f(0) by (20), since 01-A'f (0) = 0 – {log (1 + AE')} 09-1 f'(o'), where f'0')=0'A'f (0). Repeating this n-1 times we get {log (1 + AE')} 0" f(0) = (-1)^-1 {log (1+AE')} Ofn-(0') - E' fn-(0')= [(x + 1) A (x + 1) A ... f (x + 1)]x=0. This transformation has been given because it leads to a remarkable expression due to Bauer (Crelle, LvIII. 292) for Bernoulli's numbers. Denote by A' the operating factor (x + 1) A, and write 1 for f (x) and 2n +1 for n, and we obtain from (18) C 02n+1 B.xn-1 = (-1)"-l{log (1 + AE')} 0 +]...(21.) + Factorial Coefficients. 10. A series of numbers of great importance are those which form the coefficients of the powers of x when cl") is expanded in powers of x. These usually go by the name of factorial coefficients. It is evident by Maclaurin's Theorem that the coefficient DO() of 3* in the expansion of com) is *. Although it is not к n Comparing (22) page 25, and (25) page 26, we see that is the coefficient of A" in the expansion of {log (1+A)}«. That this is the case is B. F. D. 8 |