Conditional Convergence of Infinite Products

Transcription

1 Coditioal Covegece of Ifiite Poducts William F. Tech Ameica Mathematical Mothly ), I this aticle we evisit the classical subject of ifiite poducts. Fo stadad defiitios ad theoems o this subject see [1] o almost ay textbook o complex aalysis. We will estate pats of this mateial equied to set the stage fo ou esults, as follows. The ifiite poduct P = 1+a ) of complex umbes is said to covege if thee is a itege N such that 1 +a 0 fo N ad lim m=n 1 + a m ) is fiite ad ozeo. This occus if ad oly if the seies m=n log1+a m) coveges. We say that P coveges absolutely if 1+ a ) coveges. If P coveges absolutely the P coveges, but the covese is false. The followig theoem [1, p. 223] settles the questio of absolute covegece of ifiite poducts. Theoem 1 The ifiite poduct 1 + a ) coveges absolutely if ad oly if a <. If P coveges but 1 + a ) does ot, the we say that P coveges coditioally. Coditioal covegece of a does ot imply coditioal covegece of P. The followig theoem [1, p. 225] seems to be the oly geeal esult alog these lies, at least i the textbook liteatue. Theoem 2 If a 2 < the a ad 1+a ) covege o divege togethe. Hee we offe some othe esults coceig covegece of ifiite poducts. Because of Theoem 1, these esults ae of iteest oly i the case whee a =. Theoem 3 If thee is a sequece { } such that lim = 1 1) ad 1 + a ) +1 <, 2) the 1 + a ) coveges. 1

2 Poof: Let g = 1 + a ) +1. The g < 3) fom 2), so lim g = 0 ad theefoe lim a = 0 by 1). Choose N so that, 1 + a ad 1 + g / +1 ae ozeo if N. Now defie p N 1 = 1 ad p = 1 + a m ), N. m=n If N the 1 + a = p /p 1, so g = p /p 1 ) +1, ad theefoe p = +1 p g / +1 )/, which implies that p = +1 N 1 + g m / m+1 ). 4) m=n Sice 1) ad 3) imply that g m / m+1 <, Theoem 1 implies that the ifiite poduct Q = 1 + g m / m+1 ) m=n coveges; moeove Q 0 because 1 + g m / m+1 0 if m N. Now 1) ad 4) imply that lim p = Q/ N is fiite ad ozeo. To apply this theoem we must exhibit a sequece { } that will eable us to obtai esults eve if a =. The followig theoem povides a way to do this. Theoem 4 Suppose that fo some positive itege q the sequeces a k) = ae all defied, ad m= The 1 + a ) coveges. Poof: Defie a m a k 1) m, k = 1,..., q with a 0) m = 1), k) = 1 + a a q) <. 5) k 1) j a j), 1 k q. j=1 We show by fiite iductio o k that k) 1 + a ) k) +1 = 1)k a a k) 6) 2

3 fo 1 k q. Sice lim q) 5) ad Theoem 3 with = q) Sice 1) = 1 a 1) = 1 we ca the set k = q ad coclude fom that 1 + a ) coveges. the left side of 6) with k = 1 is 1 a 1) )1 + a ) 1 a 1) +1 ) = a a 1) a a 1) + a1) +1 = a a 1), sice a 1) +1 + a = a 1). This poves 6) fo k = 1. Now suppose that 6) holds if 1 k < q 1. Sice k) 1) k a k+1), 6) implies that k+1) Theefoe + 1) k a k+1) ) = k+1) a ) k+1) +1 1) k a k+1) +1 = 1) k a a k). k+1) 1 + a ) k+1) +1 = 1) k a a k) ak+1) sice a k+1) +1 + a a k) = 1) k+1) a a k+1), a a k+1) = a k+1). This completes the iductio. ) + a k+1) +1 We ow pepae fo a specific applicatio of Theoem 4. Hecefoth is the fowad diffeece opeato; thus, if {g m } is a sequece, the g m = g m+1 g m, while if G is a fuctio of the cotiuous vaiable x the Gx) = Gx + 1) Gx). Highe ode fowad diffeeces ae defied iductively; thus, if 2 is a itege, the g m = g m+1 g m = A simila defiitio yields Gx). ) 1) g m+. Lemma 1 Suppose that t is a eal umbe, ot a itegal multiple of 2π, ad {g m } is a sequece such that lim m g m = 0 ad g m < 7) fo some positive itege. The g m e imt coveges ad [ ] g m e imt = 1 e it ) A s g s + e it g m )e imt, 8) s=0 whee ) A s = 1) m s e imt, 0 s 1. 9) m s m=s 3

4 Poof: Suppose that M > 2 ad let Sice we have S M = = M g m S M = 1 e it ) 1 e it ) e imt = 1) 1) ) M+ m= M g m e imt. 10) ) 1) e im+)t, ) e im+)t = g m e imt. ) M 1) g m e im+)t Revesig the ode of summatio i the last sum yields m ) M ) S M = 1) )g m e imt + 1) )g m m= M+ ) + 1) )g m e imt. m=m+1 =m M e imt Sice lim m g m = 0 the last sum o the ight coveges to 0 as M. The secod sum o the ight is M m= M g m )e imt = e it g m )e imt, which coveges as M because of 7). Theefoe lim S m ) M = S 1) )g m e imt + e it g m ) e imt, M which ca also be witte as S = A s g s + e it g m )e imt, s=0 with A s as i 9). This ad 10) imply 8). Hecefoth we wite Gx) = Ox α ) to idicate that x α Gx) emais bouded as x. 4

5 Defiitio 1 Let F α be the set of ifiitely diffeetiable fuctios F o [1, ) such that F ) x) = Ox α ), = 0, 1, ) Fo example, let Fx) = u γ x), whee u is a atioal fuctio with positive values o [1, ) ad a zeo of ode p > 0 at ; the F satisfies 11) with α = pγ. To see this, we fist ecall that if f = fu) ad u = ux), the fomula of Faa di Buo [2] fo the deivatives of a composite fuctio says that d dx fux)) = =1 d du fu)! 1!! ) u 1 ) u 2 u )), 12) 1! 2!! whee the pime deotes diffeetiatio with espect to x. We ae assumig hee that the deivatives o the ight of 12) exist. Hee u,..., u ) ae evaluated at x, ad is ove all patitios of as a sum of oegative iteges, such that Applyig 12) with fu) = u γ yields =, 13) =. 14) F ) x) = =1γ) ) u γ x)! 1!! u x) 1! ) 1 u x) 2! ) 2 ) u )x),! whee γ) ) = γγ 1) γ + 1). Sice u l) x) = Ox p l ), it follows that whee u γ x))u x)) 1 u x)) 2 u ) x)) = Ox λ ), λ = pγ ) + p + 1) 1 + p + 2) p + ) = pγ + because of 13) ad 14). This veifies 11) with α = pγ. Fo ou puposes it is impotat to ote that F α is a vecto space ove the complex umbes. Moeove, if F i F αi, i = 1, 2, the F 1 F 2 F α1+α 2. Lemma 2 If F F α the Poof: We show that Fx) = Ox α ), = 0, 1, 2,.... Fx) K max x<ξ<x+ F ) ξ), 15) 5

6 whee K is a costat idepedet of F. Sice F ) x) = Ox α ) this implies the coclusio. To veify 15), we ote that if x > 1 ad > 0 the Taylo s theoem implies that Fx + ) = F m) x) m! m + F ) ξ ),! whee x < ξ < x +. Sice Fx) = 1) ) Fx + ), it follows that F m) ) x) Fx) = 1) ) m + 1 ) 1) F ) ξ ). m!! Sice 1) ) m = 0 fo m = 0,..., 1, we ca ow ife 15) with K = ) ) /!. Lemma 3 Suppose that F F α. Let be a fixed positive itege ad let t be a eal umbe, ot a itegal multiple of 2π. The Fm)e imt = G)e it + O α +1 ), m= whee G F α ad G depeds upo ). Poof: We wite Fm)e imt = e it F + m)e imt. 16) m= Fom Lemma 2, F + m) = O + m) α ); that is, thee is a costat A such that F + m) < A + m) α if + m > 0. Theefoe, if > 2, F + m) < A = A m) α < A +m +m 1 dx x + α) = O α +1 ). dx x + α) Applyig Lemma 1 specifically, 8)) with g m = F + m) ad fixed shows that F + m)e imt = G) + O α +1 ) with Gx) = 1 e it ) A s Fx + s), 6 s=0

7 so G F α. Now 16) implies the coclusio. The followig theoem shows that Theoem 4 has otivial applicatios fo evey positive itege q. Theoem 5 Suppose that a = f)e iθ, = 1, 2, 3,..., 17) whee f F γ fo some γ 0, 1], ad let q be the smallest itege such that q + 1)γ > 1. 18) The the ifiite poduct P = 1 + a ) coveges if θ is ot of the fom 2kπ/ with k a itege ad {1,..., q}. Poof: We show by fiite iductio o p that if p = 1,..., q the a a p) = f p)e ip+1)θ + O p+1)γ q+p ) 19) whee f p F p+1)γ. I paticula, 19) with p = q implies that a a q) = O q+1)γ ), so 18) implies 5) ad P coveges, by Theoem 4. Fom 17) ad Lemma 3 with t = θ, F = f, α = γ, ad = q, a 1) = fm)e imθ = G 1 )e iθ + O γ q+1 ), m= with G 1 F γ. Theefoe a a 1) = f)e iθ G 1 )e iθ + O γ q+1 ) ). Sice f F γ, this ca be ewitte as a a 1) = f 1 )e 2iθ + O 2γ q+1 ), with f 1 = fg 1 F 2γ. This establishes 19) with p = 1, so we ae fiished if q = 1. Now suppose that q > 1 ad 19) holds if 1 p < q. Sice p + 1)θ is by assumptio ot a itegal multiple of 2π, Lemma 3 with t = p + 1)θ, F = f p, α = p + 1)γ, ad = q p implies that f p m)e ip+1)mθ = G p )e ip+1)θ + O p+1)γ q+p+1 ), m= whee G p F p+1)γ. This ad 19) imply that so a p+1) m= a m a p) m = G p )e ip+1)θ + O p+1)γ q+p+1 ), ) a a p+1) = f)e iθ G p )e ip+1)θ + O p+1)γ q+p+1 ). Sice f F γ, this ca be ewitte as a a p+1) = f p+1 )e ip+2)θ + O p+2)γ q+p+1 ), with f p+1 = fg p F p+2)γ. This completes the iductio. 7

8 Coollay 1 Suppose that {a } is as defied i Theoem 5. The the ifiite poduct 1 + a ) coveges if θ is ot a atioal multiple of 2π. Coollay 2 Suppose that α > 0 ad R is a atioal fuctio such that Rx) > 0 o [N, ) N = itege) ad lim Rx) = 0. The the ifiite poduct =N 1 + R))α e iθ ) coveges if θ is ot a atioal multiple of 2π. Coollay 3 The ifiite poduct 1 + α e iθ) coveges if α > 0 ad θ is ot a atioal multiple of 2π. REFERENCES 1. K. Kopp, Theoy ad Applicatio of Ifiite Seies, Hafe Publishig Compay, New Yok, Ch.-J. de La Vallee Poussi, Cous d aalyse ifiitesimale, Vol. 1, 12th Ed., Libaie Uivesitaie Louvai, Gauthie-Villas, Pais, Lake Dive West, Divide, CO wtech@tiity.edu 8