On Some Basic Properties of Geometric Real Sequences

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1 On Some Basic Properties of eometric Rea Sequences Khirod Boruah Research Schoar, Department of Mathematics, Rajiv andhi University Rono His, Doimukh , Arunacha Pradesh, India Abstract Objective of this paper is to introduce geometric rea sequence and discuss their basic properties. Keywords Non-Newtonian Cacuus, geometric Arithmetic, geometric integers, geometric rea numbers. 1. INTRODUCTION In mid-seventeenth century, Newton and Leibniz created cassica cacuus considering the deviations by difference, i.e. as (x + ) f(x). In 1967 Michae rossman and Robert Katz created the first system of non-newtonian cacuus [10] and is caed Mutipicative Cacuus or exponentia cacuus considering the deviation by ratio, i.e. as f(x+a). The operations of mutipicative cacuus are caed as mutipicative derivative f(a) and mutipicative integra. In 1970 they had created infinite famiy of non-newtonian cacui, each of which are different from the cassica cacuus of Newton and Leibniz. After rossman and Katz s pioneering creation of non-newtonian cacuus, many researches have been deveoping the fied of non-newtonian cacuus and its appications. We refer rossman and Katz [10], Staney [24], Bashirov et a. [2, 3], rossman [9], K. Boruah and B. Hazarika [13, 14, 15, 16, 17, 18] for eements of mutipicative cacuus and its appications. An extension of mutipicative cacuus to functions of compex variabes is handed in Bashirov and Riza [1], Uzer [27],Bashirov et a. [3], Çakmak and Başar[5], Tekin and Başar [25], Türkmen and Başar [26]. Kadak and Özük studied the generaized Runge-Kutta method with respect to non-newtonian cacuus. Kadak et a [28] studied certain new types of sequence spaces over the Non- Newtonian Compex Fied. eometric cacuus is aso a type of non-newtonian cacuus. It provides differentiation and integration toos based on mutipication instead of addition. Every property in Newtonian cacuus has an anaog in mutipicative cacuus. eneray speaking mutipicative cacuus is a methodoogy that aows one to have a different ook at probems which can be investigated via cacuus. In some cases, for exampe for growth reated probems, the use of mutipicative cacuus is advocated instead of a traditiona Newtonian one. The main aim of this paper is to discuss about the basic properties of geometric sequences. Before we estabish new resuts, we reca the construction of arithmetics generated by different generators and the geometric arithmetic, which is the keyword of the whoe artice. 2. α ENERATOR AND EOMETRIC REAL FIELD A generator is a one-to-one function whose domain is the set of rea numbers, R and range is a set B R. For exampe, the identity function I and the exponentia function exp are generators. We consider any generator α with ream i.e. domain, say, A and range B, by α aritmetic, we mean the arithmetic whose operations and ordering reation are defined as foows: forx, y A, where A is a domain of the function α. α addition x+y = α[α 1 (x) + α 1 (y)] α subtraction x y = α[α 1 (x) α 1 (y)] α mutipication x y = α[α 1 (x) α 1 (y)] α division x/y = α[α 1 (x)/α 1 (y)] α order x < y α 1 x < α 1 y. It is to be noted that each generator generates exacty one arithmetic and each arithmetic is generated by exacty one generator. For exampe, if we choose the exponentia function exp as an α generator defined by α(x) = e z for x R and hence α 1 (x) = nx, then α aritmetic turns out to geometric arithmetic as foows: ISSN: Page 111

2 geometric addition x y = α[α 1 (x) + α 1 (y)] = e (nx +ny ) = x. y geometric subtraction x y = α[α 1 (x) α 1 (y)] = e (nx ny ) = x y, y 0 geometric mutipication x y = α[α 1 (x) α 1 (y)] = e (nx ny ) = x ny geometric division x y = α[α 1 (x)/α 1 (y)] = e (nx ny) = xn y, y 1. Simiary, the identity function generates cassica arithmetic. It is obvious that n(x) < n(y) if x < y for x, y R +. That is, x < y α 1 (x) < α 1 (y). So, without oss of generaity, we use x < y instead of the geometric order x < y. C. Türkmen and F. Başardefined the sets of geometric integers, geometric rea numbers and geometric compex numbers Z(), R() and C(), respectivey, as foows: Z() = {e x : x Z} R() = {e x : x R} = R^ + {0} C() = {e z : z C} = C {0}. If we take extended rea number ine, then R() = [0, ]. (R(),, )is a fied with geometric zero 1 and geometric identity e, since (1). (R(), ) is a geometric additive Abeian group with geometric zero 1, (2). (R() {1}, ) is a geometric mutipicative Abeian group with geometric identity e, (3). is distributive over. But (C(),, ) is not a fied, however, geometric binary operation is not associative in C(). For, we take x = e 1/4, y = e 4 and z = e (1+iπ /2) = ie. Then (x y) z = e z = z = ie Butx (y z) = x e 4 = e. Let us define geometric positive rea numbers and geometric negative rea numbers as foows: R + = x R : x > 1 R = x R : x < Some Usefu Reations between eometric Operations and Ordinary Arithmetic Operations: For a x, y R() x y = xy x y = x/y x y = x ny = y nx 1 a b y = a b ny a n y a y = bn y = b y a b c d = a b n c d = b a n c d = b a n d c = b a d c x y or x y = x 1 n y, y 1 x 2 = x x = x nx x p = x np 1 x x = e (nx) 1 2 ISSN: Page 112

3 x 1 = e 1 og x x e = x and x 1 = x e n x = x x..... (upto n numberof x) = x n Thus x 1. x 2 = x e y = e y x y = x y x y x y x y = x y x y x y x = x, if x > 1 1, if x = 1 1, x if 0 < x < x y = y x, i. e. in short x y = y x. Further e x = e x hods for a x Z +. Thus the set of a geometric integers turns out to the foowing: Z() = {..., e 3, e 2, e 1, e 0, e 1, e 2, e 3,... } = {..., e 3, e 2, e, 1, e, e 2, e 3,... }. 3. MAIN RESULTS: EOMETRIC REAL SEQUENCE A function whose domain is the set N of natura numbers and range a set of geometric rea numbers is caed a geometric rea sequence. We denote geometric rea sequence as S: N R(). Since, we wi discuss about geometric rea sequences ony, we sha use the term geometric sequence to denote geometric rea sequence. A geometric sequence wi be denoted by {S n } or { S 1, S 2, S 3,..., S n,... } where n N. Numbers S 1, S 2, S 3,... wi be caed first term, second term, third term, of the sequence, respectivey. As we as the ordinary rea sequence, a the terms of geometric sequence wi be treated as distinct terms athough some terms are equa in different situations. Exampe of geometric sequence: 1. {S n } = {e ( 1)n, n N}. 2. {S n } = {e n, n N}. 3. {S n } = {e 1+( 1)n, n N}. 3.1 Bounds of eometric Sequence: It is obvious that the set R() = R + {0} of geometric rea numbers is bounded beow. So, geometric sequences are aways bounded beow. A geometric sequence is said to be bounded if it is bounded above. That is, a sequence {S n } is bounded if there exists a rea number K such that S n K, n N Convergence of eometric Sequences: Definition:3.1. A geometric sequence {S n } is said to be convergent to a rea number > 0 if for ε > 1, there exists a positive integer m depending on ε such that S n < ε, for a n m.in ordinary sense ε < S n < ε for a n m. In other word, terms of the sequence approach the vaue by ratio as n becomes arger and arger. Symboicay, we write x a S n = or S n ISSN: Page 113

4 and we say that the sequence geometricay converges to or -convergent to. Since, ε < S n < ε, for a n m, hence infinite number of terms ie to the eft of ε and infinite number of terms ie to the right of ε. Theorem 3.1.-convergent sequence are bounded. Proof. Let the geometric sequence {S n } converge to. Then for given ε > 1, there exists a positive integer m such that < S ε n < ε, for a n m. Now, since ε > 1 and > 0, hence and ε are finite numbers. Let ε L = min{ ε, S 1, S 2,.., S m 1 }. U = max{ε, S 1, S 2,.., S m 1 }. Then, L S n U, for a n m. Hence the sequence is bounded. Theorem 3.2.Boundedness need not impy geometric convergence. Proof. Let us consider the sequence {S n }, where S n = e ( 1)n, n N such thats n. Then for ε = e, m N, such that ε < S n < ε n m e < e( 1)n < e e < e( 1)2m < e and e < 1 e( 1)2m < e < e < e e and e < e 1 < e e 2 < 1 < and < 1 < e2 1 < and < 1 which is impossibe at the same time. Hence, the sequence is not convergent. Aso, it can be proved that a geometric sequence can not -converse to more than one imits Limit of a Function: According to rossman and Katz [10], geometric imit of a positive vaued function defined in a positive interva is same to the ordinary imit. We have defined -imit of a function with the hep of geometric arithmetic in as foows: A function f, which is positive in a given positive interva, is said to tend to the imit > 0 as x tends to a R, if, corresponding to any arbitrariy chosen number ε > 1, however sma(but greater than 1), there exists a positive number δ > 1, such that 1 < f(x) < ε for a vaues of x for which 1 < x a < δ. We write f(x) = x a Here, or f(x). x a < δ x a < δ 1 δ < x a < δ Simiary, f(x) < ε < f(x) < ε. ε a < x < aδ. δ Thus, in ordinary sense, f(x) means that for any given positive rea number ε > 1,no matter however coser to 1, a finite number δ > 1 such that f(x) ], ε[ for every x ] a, aδ[.it is to be noted that ε δ engths of the open intervas ] a, aδ[ and ], ε[ decreases as δ and ε respectivey decreases to 1. Therefore, as ε δ ε ISSN: Page 114

5 decreases to 1, f(x) becomes coser and coser to, as we as x becomes coser and coser to a as δ decreases to 1. Hence, is aso the ordinary imit of f(x). i.e. f x f x.in other words, we can say that -imit and ordinary imit are same for bipositive functions whose functiona vaues as we as arguments are positive in the given interva. Ony difference is that in -cacuus we approach the imit geometricay, but in ordinary cacuus we approach the imit ineary. A function f is said to tend to imit as x tends to a from the eft, if for each ε > 1 (however sma), there exists δ > 1 such that f(x) < ε when a/δ < x < a.in symbos, we write f(x) = or f(a 1) =. x a Simiary, a function f is said to tend to imit as x tends to a from the right, if for each ε > 1(however sma), there exists δ > 1 such that f(x) < ε when a < x < aδ.in symbos, we then write f(x) = or f(a + 1) =. x a+ If f(x) is negative vaued in a given interva, it wi be said to tend to a imit < 0 if for ε > 1, δ > 1 such that f(x) ]ε, ε [ whenever x ] a δ, aδ[ Continuity of a Function[16]: A function f is said to be -continuous at x = a if 1. f(a) i.e., the vaue of f(x) at x = a, is a definite number, 2. the -imit of the function f(x) as x a exists and is equa to f(a). Aternativey, a function f is said to be -continuous at x = a, if for arbitrariy chosenε > 1, however sma, there exists a number δ > 1 such that f(x) f(a) < ε for a vaues of x for which, x a < δ. On comparing the above definitions of imits and continuity, we can concude that a function f is continuous at x = a if f(x) im x a f(a) = Limit Point of a eometric Sequence: A rea number ξ is said to be a -imit point of a geometric sequence {S n } if for a given ε > 1 however sma but greater than 1, S n ] ξ, ξ[ for an infinite number of vaues of n N. Thus, ξ is a imit point of the ε sequence if every neighborhood of ξ contains an infinite number of members of the sequence. It can be easiy proved that Bozano-Weierstrass Theorem is aso vaid for infinite set of geometric rea numbers and geometric sequences. Theorem 3.1 (Cauchy s genera principe of geometric convergence). A necessary and sufficient condition for the convergence of a geometric sequence {S n } is that, for ε > 1 there exists a positive integer m 0 such that S n S m < ε, n, m m 0. That is, S m +1 S m +2 S n < ε, n, m m 0 S m +1 S m +2 S n < ε. Equivaenty, we can say that the sequence {S n } is convergent if and ony if for ε > 1 there exists a positive integer m such that S n+p S n < ε, n m, p 1. Theorem3.2 If{a n } and {b n } be two convergent sequences such that ima n = a, imb n = b, then for a n N 1. im(a n b n ) = a b, 2. im(a n b n ) = a b, 3. im(a n b n ) = a b, ISSN: Page 115

6 4. im(a n b n ) = a b. Proof. (i) We have discussed that in case of bi-positive functions, geometric imit and ordinary imit are same. Since terms of a geometric sequence are aways positive, hence ima n = a ima n = a and imb n = b imb n = b.now, (ii) (iii) (iv) Proof is simiar. im(a n b n ) = im(a n b n ) = im(a n b n ) = im(a n )im(b n ) = ab. im(a n b n ) = im(a n /b n ) = im(a n /b n ) = im(a n )/im(b n ) = a/b = a b. im(a n b n ) = im (a n ) n(b n ) = im (a n ) n(b n ) = im e n(a n ).n (b n ) = e im n (a n ).n (b n ) na.nb = e = a n (b) = a b. Theorem3.3 If ima n =, such that > 0, then a 1 a 2... a n e n =. Proof. We know that, if a positive term series {a n } converges to positive number, then im a 1 a 2... a n 1 n =.Since, terms of a geometric sequence age aways positive and ima n = ima n =, hence a 1 a 2... a n e n = im a 1 a 2... a n 1/n (e n ) = im a 1 a 2... a 1/n n = im a 1 a 2... a 1/n n =. 5. ACKNOWLEDMENT It is peasure to thank Professor Bipan Hazarika, Department of Mathematics, Rajiv andhi University, Itanagar, for his constructive suggestions and corrections made for the improvement of this paper as we as eometric cacuus. 6. CONCLUSION Here we have discussed about some basic properties of geometric rea sequences. In [13] we have introduced geometric difference sequence spaces which are based on geometric arithmetic. Here we have just introduce geometric rea sequences, -imit, -continuity and their basic properties which wi be hepfu for the deveopment of the eometric Cacuus. REFERENCES [1] Bashirov, M. Riza, On Compex mutipicative differentiation, TWMS J. App. Eng. Math. 1(1)(2011), [2] A. E. Bashirov, E. Misiri, Y. Tandoǧdu, A. Özyapici, On modeing with mutipicative differentia equations, App. Math. J. Chinese Univ., 26(4)(2011), [3] A. E. Bashirov, E. M. Kurpinar, A. Özyapici, Mutipicative Cacuus and its appications, J. Math. Ana. App., 337(2008), [4] F. Başar,, Summabiity Theory and Its Appications, Bentham Science Pubishers, e-books, Monographs, xi+405 sf., Istanbu- 2012, ISBN: [5] A. F. Çakmak, F. Başar, On Cassica sequence spaces and non-newtonian cacuus, J. Inequa. App. 2012, Art. ID , 12pp. ISSN: Page 116

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