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On Infinite Products :?!t--:v.- -, View metadata, citation and similar papers at core.ac.ukbrought to you by CORE provided by K-State Research Exchange ON INFINITE PRODUCTS by ROSE KORDONOWY SHAW B. S. , Dickinson State College, 1961^ A MASTER'S REPORT submitted in partial fulfillment of the requirements for the degree MASTER OF SCIENCE Department of Mathematics KANSAS STATE UNIVERSITY" Manhattan, Kansas 1966 Approved by; :o7 U) '^ JScGV ii '3^ TABLE OP CONTENTS INTRODUCTION 1 BASIC DEFINITIONS 1 CONVERGENCE OF INFINITE PRODUCTS i; RELATING CONVERGENCE OF INFINITE PRODUCTS TO CONVERGENCE OF INFINITE SERIES 9 ABSOLUTE CONVERGENCE 21 THE ASSOCIATE LOGARITHMIC SERIES 2i|. UNIFORM CONVERGENCE 28 DEVELOPMENT OF TRIGONOMETRIC FUNCTIONS AS INFINITE PRODUCTS 3i| ACKNOWLEDGMENT l^^ BIBLIOGRAPHY l^]^ INTRODUCTION Many functions can be represented in many different ways. They can be represented by Fourier series, orthogonal poly- nomials, infinite series, and infinite products. Infinite products will be considered as the background for the develop- ment of expansion of functions. The infinite product repre- sentations of sin X and cos x will be considered. The theory of infinite products is very closely related to that of infinite series; so closely, in fact, that the test for convergence of such products will actually be reduced to the test for the convergence of certain infinite series. The object of this paper is to begin with the basic defini- tion of an infinite product and to take the study of infinite products to the point where a basic understanding of infinite products and how they can be handled is reached. BASIC DEFINITIONS An infinite product is of the form . 8;^ 82 . a^ . ., where a^ is a complex number, i = 1, 2, ..., n, ..., and is denoted by oc "" ~ 7T 8k ^1 ^2 • • • sjj. k=l The partial products p-^ are defined in the following manner: . Pi = n _ P2 = a^^ . 82 - • Pk ^1 ^2 . a^^ = where k 1, 2, 3, . , With these preliminary comments on the form of infinite products, convergence, divergence, and oscillation of an infinite product will be defined'. oo Definition 1. The infinite product Tf a,^ converges if and k=i only if there exists a K such that ' K+g lira1 .^r r. " ,^ J r^ g-*oo £k ^ K-1 If the infinite product converges, its value is A Tf a,^. k=l oo Definition 2. An infinite product 7T a;^ is said to be k=l divergent if A in definition 1 is equal to zero for all K or when A is infinite. Definition 3- An infinite product is oscillating when A does not approach any definite limit. An oscillating product is not convergent. The following examples will be presented in an attempt to clarify the meanings of definitions 1, 2, and 3. Example 1. The . infinite product . 2 .-2 ... where = and = ^2k-l a2k 2 for k = 1, 2, . is divergent because it contains an infinite number of zeros. i.e., there does not exist a K such that g->oo ,TC B^ = A ^ 0. k=K The infinite product oo 1 - = • . Tf 1 1/2 1/3 . 1/k . k=l k / is divergent because, although it does not contain any zeros, • lim _ . k-»co Pk - 0- The infinite product, k=1.2.3...k...,is divergent because limit p^ = oo . k->oo oo The infinite product IT (-1)^ = -1 . i . -i ... (-i)k _ k=l is oscillating because the product does not approach any defi- oo nite limit. Consider the partial products of Tf (-1) . k=l Pi = -1 Pg = -1 P3 = 1 It is seen that the infinite product is not approaching any definite limit. The infinite product O.O-l.l.l.i... where a^ = for k = 1, 2 and ^i, = 1 for k = 3, if, 5, ... ^ -u limit irr-^ IS convergent because „ .„ 7( a, = 1. The value of the 2 Infinite product is 1 . JC Qr^ - . k=l It can be seen from the above definitions and examples that a convergent infinite product can contain a finite number of oo aj_'s equal to zero. If a, = TT ^ 0, it is not known whether a k=l finite number of a^^'s are equal to zero and the infinite product is convergent or if all a^'s are not equal to zero and oo TT ai^^ = and the product is divergent. It can also be seen k=l that oscillating infinite products occur when some a- «= 0. To simplify the theory to be presented, it will be assumed that a-j_ > for every i = 1, 2, . CONVERGENCE OP INFINITE PRODUCTS ". At this point, the characteristics of convergent infinite products vjill be considered. Theorem la. The sequence of the factors in a convergent infinite product always tends to approach one. The proof is as follows: Pk-1 = ^1 • ^2 ••• ^k-l Pk = ^1 • ^2 • • • sj^ since p^.-^ approaches A and approaches p^ A and A f^ Pk _ ai • 32 • •• ®k-l • ^k ' = Sk -> 1 • • • • Pk-1 ^1 ^2 • Sk-1 The following theorem is closely related to theorem la, and hence will be ten^ied theorem lb. CO The orem lb. For the infinite product Tf a^ to converge. k=l it is necessary and =* sufficient that for every £ , a K > exists such that for every k =• K and for every integer r ^ 1 ~ " . ^k+1 ^k+2 • • • ®k+r e The proof of theorem lb is as follows. CO If Tf a^ is convergent, then the sequence of the factors k=l in a convergent infinite product always tends to approach one. Let Pk+r ^k = • Pk Because p^ converges to a limit P, lim _ k-^cx) ^k ~ ' which implies that " ^ ^ ^ ^o^ k => K. I ^k I But = ^k ^k+1 \+2 ' ' ' \+r and hence ~ '• ^k+1 • • • I ^k+2 \+r 1 6 and the necessity part of the proof is complete. If Pk+r ^k Pk and if the condition - 6" ^k 1 is satisfied, then 1 - e ^k e + and Pk+r ^k ^ 1 Pk and Pk+r-> Pk which implies that for every €" > 0, Pk+r - Pk e . This is the Cauchy condition for the convergence of a sequence. Therefore CO is convergent, and the theorem is proved. This theorem gives the general convergence principle. On the "basis of this theorem = = (let r 1 and k + 1 n) , it is necessary that the ^^^ a = 1 n->oo n -^ • Therefore set the factors of the product equal to (1 + u ) and the following form of the infinite product arises. oo TT (1 + uj = (1 + + ui)(l U2) ... (1 + uj ... n=l " Because the assumption was made that every a- > 0, every (1 + u^) =• 0, and \x^ =• -1. The partial product P^ is defined as P^ = (1 + U]_)(l + U2) ... (1 + Uj^) for n = 1, 2, 3, ... 00 The infinite product 7T (1 + u^^) is convergent if and n=l a such that ~ -^ only if there exists finite ? ^ ^^^q -^n exists; otherwise the infinite product is said to be divergent. 00 If the infinite product Tf (1 + a„) is convergent, then n=l by the general condition of convergence of a sequence P - P , n > N, = 1, 2, ... n+p n C p 3, , and also there exists a g such that Pn for every n, where g is a positive constant. The condition n * g implies that ^n ^ *-* -^°^ every n for a convergent product. Sufficient background has been developed so that the fol- lowing theorem for convergence can be considered. Theorem 2. A necessary and sufficient condition for the convergence of the infinite product CX) TT (1 + u^) n' n=l is that, for any arbitrary small 6 =* 0, an N > can be found such that for n S n^ "^ pn+p n where p is a positive integer. 8 Suppose first that the product is convergent, then since n ^n+p " ^n 6 can be divided by n getting n+p 1 i = e. n The condition is therefore necessary. If now the condition ^n+p n. is satisfied, for any assigned C , the inequality can be written u^^2) -^ (1 + %+i)(l + ••• (1 ^n+p) - 1^ ^ for n > N, and for every p, and by theorem lb the infinite product is convergent. If in the above theorem p is set equal to 1, P.n+1 = 1 + un+1 n and u £ n N, n+1 , > which gives the next theorem. Theorem 3- A necessary condition for convergence of the ^" infinite product TT (l + u„) is that J"™ = 0. n=l " n->oo This condition is not in general a sufficient condition. 1 This is shown by the example of Uj^ = + - , for n n 1 111 1 n (1 + -) = (1 + -)(i + -)(i + -) ... (1 + -) i=l i 12 3 3 i; 5 n+1 = n + 1 2 3 li^ n and n 1 11 1 ff - = (1 - -)(1 - -) ... (1 - -) 1=2 i 2 3 n 123i| n-11 2 3 i| 5 n n so that both infinite products are divergent, since for the lim u first P^ -» oo and for the second P^ "^ ^' ^^'^ y®^ n-^c'^^oo ~ ^ for both. RELATING CONVERGENCE OF INFINITE PRODUCTS TO CONVERGENCE OP INFINITE SERIES Definitions and theorems on the convergence of infinite products have been given. The next step is to relate the con- vergence of infinite products to the convergence of an infinite series. To develop an equivalent series, consider the partial product P^ with the additional condition that Pq = 1 and the recursion formula Pi = (1 + Ui)Pi_i = Pi-1 + ^i Pi-1 • It follows that ^n = Pn-1 + % Pn-1 = ^n-2 + ^n-1 Pn-2 + ^n ^n-l = Pn-3 ^ ^n-2 Pn-3 "^ ^n-1 ^n-2 + ^n ^n-l Therefore (I) ?n = 1 ^ • 1=1L Vi-1 If the infinite product converges, then exists, n"^^"Ji)* ^n and 10 hence the series 1=1 is convergent with sum equal to the value of the product.
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