< Let Vn+1 <vnk where k is a positive quantity <l. Then raising the product V,V,V3... Ur... to the power 1 – k, we have V, • v, -* . Vz. V, -* . Vz. v in which the factors are alternately greater and less than 1; and also the pairs of factors V1.0,-%; 0,-*. v, ; v, -*.v3, &c., are alternately greater and less than 1. (Cf. Art. 450.) 519. If v, always decreases as n increases and is always positive, then according as the series whose general term is vn is convergent or divergent, so also is the product whose general factor is For let log (1 + vn) = Un, so that 1 + Vn = = e un em – 1 Un Un 1 + + Un 2 3 Un + + Un n nor can This series cannot be zero or infinity for any finite value of Un log (1 + on) Vn (including zero). That is, neither log (1 + vn) Vn be infinite for any value of n. .. (by Art. 453) according as the series whose general term is Vm is convergent or divergent, so is the series whose general term is log (1 + vn), and so also is the product whose general factor is 1 + Uno The same argument applies to the product whose general factor is 1 – Vn, where vn <l; which of course cannot be infinite but may be zero. Example. The limit of the coefficient of an in the expansion of (1+x)m is zero or infinity according as m is greater or less than – 1. For the coefficient of x is m +1 m+1 2 3 This product is always divergent, because the sum 1+3+3+... is divergent. Hence the product is zero or infinity according as m +1 is positive or negative. (See Art. 462.) 520. It was shown in Arts. 416, 418 that } (ac" + x-n) is the same function of } (x + 2c-?) as cos na is of cos a: and that xn - x-n is the same function of 1 (2+) as is of cos a. sin na sin a Put x = ea; then } (x+2-1) = 1 (ea +e-a)= cosh a; } (x – 2-4) sinh a; 3 (+c") = cosh a; } (e” – ch”)=sinh a. Thus cosh na is the same function of cosh a as cos na is of Hence the second formulæ of Arts. 419, 420, 421 become cosh no 3п 2n-1 -COS 2n 1 sinh mà 2n-1 sinh 0 ( cosh 0 – cos cosh 0 T (cosh T n - COS n n ) 3.) ... (cosh -), a =)} cosh no - cos na= COS n cosh 0 – 2n-2 a + n 521. T'o resolve cos 0 and cosh O into factors. 37 2n-1 2n-1 ( cos a – COS 2n 2n -), ( T 2n-1 ( cosh a - COS 2n) (cosh cosh a 3T COS 2n) cosh a 2n 1 COS T 212 Now cos (2n – 1) 7/2n = cos λπ/2n. 2η –λ 2n 2na COS 2n : a – ) ) (cosh ); 20-^ = sin cosh a λπ ? + sinh? a. 2n T sino a sino a 2 sino a a)... (sin" a) (sino a) (sino a) a). T .. if n is even, multiplying first and last factors &c., we have 37T n-1 2n-1 (sin 2n 2n 2n 37 n-1 2n-1 (sin + sinh? a. + sinh a + sinh? a. 2n 2n 2n Here put na=0. Thus cos Q and cosh 0 are equal respectively to 8: 37 n-1 1 ? ? 2n 2 a) ... (sir Here put 0= 0. Thus Divide each of the above two equations by this last. Thus sinon sinh? 6 m cosh 0 =(1+ 7 sino 37/2n/ sin" (n-1) 7/2n) In these last equations, make n infinite, while 6 remains constant and finite. Then 0/n and XT/2n become zero: and, for the ratio of the functions sin and sinh we may substitute that of the quantities O'n and do/2n themselves. Thus 1 to infinity 20 57 008 0=1-1-(3){1- 1+( } T 1 These products are convergent for all finite values of 0, because the sum of the series 2012 1 1 1 + 52 72 is convergent: the nth term being of the form (2n + 1)“, where -C=-2 <-1, G+ 5+ 6+ ... ) ) (it The first formulæ may easily be remembered, by giving to 6 37 57 the values &c. which make each side vanish. 2' 2 2 TT :. if n is odd, multiplying second and last factors &c., 1 sin na = 2n-1 sin a ( sin sin? 2n Here put 0=0. Thus 2. n=2n-1 sin?“ sin? n n T Thus Divide each of the above two equations by this last. 0 sinon 1 sin(n − 1) 7/2n sinh? 6 m 1+ 1+ ? | sino (n − 1) 7/2n 2n) n T (1 m sinh n Here make n infinite. Thus sin A=0 9=0{1-(93{1-()}(1-(39) ...to infinity. )} sinh 6 – 1+%(%)(4 {1+0)}{1+(29)} {1+ ( 3 ) } ... to infinity. 2 = + 37 -1 COS na COS a COS + λ0 n (PAT + 8)}, B cosh na ز λ=0 n These products are also convergent. The first formula may easily be remembered by giving to e the values O, T, 21 &c., which make each side vanish. 523. To resolve cos 0 – cos and cosh 0 – cos o into factors. We have 2λπ cos nß = 21-1 P = 0? {c 2λπ cos nß = 2n-1 plan-1 cosh a - cos { where P denotes the product of the factors obtained by giving to À the integral values between the assigned limits. Now cos (287/12 + B) = cos {2 (n − 1) 7/n-B}, .. if n is even, 1 = {n gives cos (287/n + B) =- cos B, and cos nß 1= -1 = 2n-1 (cos’ a - cosB) P. {co cosh na cos nß 2-1 =2n-1 (cosh2 a - cosa B) P. 2λπ {cosh a-cos -- {cos a = con (- )} food a–006 (32-->)} B{ = , (2T+B){cosh a – cos (P = B)} {cosh a B)} pin (CNT - 8) B 2λπ n TT FR 2λπ a n + sinh? a. n |