342 PROPERTIES OF EULER'S NUMBERS CHAP. Cor. 1. Euler's numbers are all positive; they continually increase in magnitude, and have infinity for their upper limit. . .. Also, and But this last constantly increases with m, and is already greater than 1, when m = 1. Hence E, <E, < E3 < from (16), we see that LT2m+1= 1 when m = L(2m)! (2/)2m+1, hence LE = ∞ . Cor. 2. Em/(2m)! ultimately decreases in a geometrical progres- sion whose common ratio is 4/π3. the secant series is 0. = * The remarkable summations involved in the formulæ (1), (2), (3) were discovered independently by John Bernoulli (see Op., t. iv., p. 10), and by Euler (Comm. Ac. Petrop., 1740). XXX SUMS OF CERTAIN SERIES 343 Inasmuch as we have independent means of calculating the numbers Bm and Em, the above formulæ enable us to sum the various series involved. It does not appear that the series σ2m+1 can be expressed by means of Bm or Em; but Euler has calculated (to 16 decimal places) the numerical values of σ2m+1 in a number of cases, by means of Maclaurin's formula for approximate summation.* As the values of om are often useful for purposes of verification, we give here a few of Euler's results. It must not be forgotten that the formulæ involving for σm are accurate when m is even; but only approximations when m is odd. § 16.] From the formulæ of §§ 6 and 7, we get, by taking logarithms, since the double series arising from the expansions of the logarithms is obviously convergent, provided mod <. If we express om by means of Bernoulli's numbers, (1) may The corresponding formule for cos are log cos 0 = (22m – 1)σm02m/m2 m CHAP. (2); (2'). The like formula for log tan 6, log cot 0, log sinh u, log cosh u, &c., can be derived at once from the above. If a table of the values of σm or of Bm be not at hand, the first few may be obtained by expanding log (sin 0/0), that is, log (1 - 0/3! +0/5! - . . .), and comparing with the series -20m02m/mr2m For example, we thus find at once that $ 17.] Before leaving this part of the subject, we shall give an elementary proof of a theorem of great practical importance which was originally given by Stirling in his Methodus Differentialis (1730). When n is very great, n! approaches equality with √(2nπ)(n/e)"; or, more accurately, when n is a large number, we have n!= √(2πn)(n/e)" exp {1/12n + 0} where 1/24n < 0 < 1/24n(n − 1). Since log {n/(n - 1)} = log (1 - 1/n), we have log (1), We can deprive this expansion of its second term by multiplying by n - 1. We thus get Hence, taking the exponential of both sides, and writing suc |