Bounded Infinite Sequences/Functions : Orders of Infinity

Transcription

1 Bounded Infinite Sequences/Functions : Orders of Infinity by Garimella Ramamurthy Report No: IIIT/TR/2009/247 Centre for Security, Theory and Algorithms International Institute of Information Technology Hyderabad , INDIA April 2009

2 BOUNDED INFINITE SEQUENCES / FUNCTIONS : ORDERS OF INFINITY Garimella Rama Murthy, Associate Professor, International Institute of Information Technology Hyderabad, Gachibowli, HYDERABAD 32, AP, INDIA ABSTRACT In this research paper, we consider the class of divergent infinite series and provide a simple approach to order them based on a concept called, convergence exponent. Some simple Lemmas are proved. The results are generalized to non integrable functions. 1. Introduction: Ever since the dawn of civilization, homosapien living machine expressed curiosity in the concept of counting. Early research efforts in basic number theory distinguished prime numbers. Euclid provided an elegant proof that the cardinality of set of prime numbers is infinite. For quite a few centuries, the concept of counting was considered to be self evident and obvious. Efforts to place calculus on a sound mathematical footing led to detailed research efforts related to the set of real numbers. Infinite sequences of real numbers were associated with formal ideas/concepts such as limit, lim inf, lim sup etc. Mathematicians did not pay much attention to the notions of countable and uncountable sets. Cantor was the first mathematician to formally define the concept of a countable / uncountable set. Also advancing the basic ideas, the concepts of ordinal as well as cardinal numbers were proposed. Mathematicians proposed the concept of an infinite series associated with an infinite sequence of real / complex numbers. It was thought that a necessary and sufficient condition for an infinite series to converge is that the terms approach zero. Bernoulli demonstrated that this conjecture is false by showing that the terms of a harmonic series approach zero but the series diverges. Thus, the condition that the terms should approach zero ( for convergence of the associated infinite series ) is necessary but not sufficient. Various tests for convergence of infinite series were designed by mathematicians. ****In part the paper appeared in G. Rama Murthy, Weakly Short Memory Stochastic Processes: Signal Processing Perspectives, Proceedings of International Conference on Frontiers of Interface between Statistics and Sciences, December 20, 2009 to January 02, 2010, Pages

3 The conventional definition of convergence of an infinite series is that the partial sums i.e. converge to a finite limit. Researchers gave more general definitions of convergence using different criteria. For instance, the finiteness of limit of arithmetic mean of terms i.e. is utilized as a criterion for convergence. This line of reasoning has been utilized ( in association with divergent series in the traditional sense ) to develop a body of research literature [Kno]. The author, while understanding the short memory stochastic processes became interested in finer classification of long memory stochastic processes [Rama1]. He realized that the ideas apply equally well to divergent infinite series. A methodical logical thinking based on those ideas resulted in this research paper. This research paper is organized as follows. In section 2, an interesting class of divergent infinite series are considered and an approach to order them is proposed. In Section 3, we consider bounded, non integrable functions and propose an approach to order them. Finally, in Section 4, ideas are proposed for generalization of the results to larger class of series and functions. 2. Divergent Infinite Series Based on Bounded Sequences : Orders of Infinity: Consider a bounded infinite sequence, Let all the terms of the sequence be bounded in magnitude by a single constant K. Define for all n. Choose k in such a way that all the terms of the sequence bounded in magnitude by one i.e. are strictly Now define as the of the infinite dimensional vector corresponding to the infinite sequence }. Let for Lemma 1 : is a strictly monotone decreasing function for increasing values of p. Thus, for integer values of p, we have that

4 Proof : Since and, basic properties of real numbers ensures that is a strictly monotone decreasing function of p. Q.E.D In view of the above Lemma, if <, then for all other values of p larger than one i.e the of the infinite dimensional vector corresponding to the infinite sequence is finite. But it can happen that the of the infinite dimensional vector is infinite. In view of these facts, the following Lemma is very interesting for the sequences satisfying the following assumption Assumption: We consider bounded infinite sequences which converge to a constant value less than one in magnitude. Lemma 2: For every infinite dimensional vector corresponding to a bounded, convergent ( converging to a value less than one in magnitude ) infinite sequence, there exists an integer p such that the is finite ( and hence for all larger values of p ). Proof: Since, for all values of n, it is evident that approaches zero as p approaches. Thus converges to zero. Consequently, there exists an integer such that < for all integer values of p larger than. Q.E.D. Note: Consider a sequence of functions such that (x) < for. Define. The results derived above can be generalized for this more general index ( associated with an infinite series ). The Lemma 2 naturally leads us to the following definition: Definition: Consider infinite series based on infinite sequences whose terms are bounded and converge to a value less than one in magnitude. is called the convergence exponent if = for all q < and for all.

5 Note: A finite convergence exponent always exists for the infinite series considered in the above definition. Note: It is well known that every monotone bounded sequence always converges. Thus, all such sequences satisfy the assumption in the above definition provided the limit is less than one. Now based on Lemma 1, Lemma 2 and the above definition, we propose an approach to order the divergent series under consideration. Definition : Consider any two infinite series with the associated convergence exponents diverges to a higher order of infinity compared to The sequence if and only if Note: The convergent / divergent infinite series which have the same convergence exponent ( unity for convergent series ) belong to the same equivalence class. Zeta Function : Motivation for Orders of Infinity: The motivation to the above discussion is based on the sequence of reasonings that led to the Riemann Zeta Function. Bernoulli s argument led to the proof that the harmonic series diverges even though the terms ( of the infinite series ) approach zero. Euler was naturally led to calculate the sum of the infinite series. Based on an ingenious argument he showed that the sum was equal to. He was able to successfully generalize the proof to sum of reciprocals of even powers of integers ( upto 26 th exponent ). From this reasoning, the following observations are made: (i) The convergence exponent of harmonic series is 2 (ii) The convergence exponent of the infinite series is 4. From the observation (ii), it is clear that this series diverges to a higher order of infinity than the one in (i) ( i.e. the harmonic series ).

6 3. Non Integrable, Bounded Functions : Orders of Infinity: The discussion in the above section naturally motivated us to consider associated with bounded functions ( except on a set of measure zero ). Specifically, let us consider a function f(.) which is bounded by a real number K in magnitude. Define g(x) = for all the domain of f(.). With out loss of generality, K can be chosen such that g(x) < 1 for all except on a set of measure zero. Consider the of the function g(.) i.e. for p > 1. The following Lemmas follow from the same reasoning as that in Section 2. Details are avoided for brevity. Lemma 3: : is a strictly monotone decreasing function for increasing values of p. Specifically, for integer values of p, we have that Hence, we necessarily have that Lemma 4: For every bounded function considered above ( i.e. bounded except on a set of measure zero ), there exists an integer such that Remark 1: Ordering of Bounded, Non Integrable Functions: As in Section 2, the utility of above Lemmas is to order the functions which are not integrable. Once again, the ordering of non integrable bounded functions is based on the definition of convergence exponent ( exactly as in the above section ). Details are avoided for brevity. One can achieve well ordering of non integrable functions. 4. Other Cases: Every infinite series is based on the associated infinite sequence. The class of infinite sequences can be classified into (i) bounded sequences and (ii) unbounded sequences.

7 Furthermore bounded sequences can be classified into (i) convergent sequences and (ii) non convergent sequences. The convergent sequences can be categorized into (i) those converging to zero and (ii) those converging to non zero value ( which is either less than unity or higher than unity. The case where the limit is equal to unity is more interesting ). Thus, the approach in Section 2 can be naturally utilized to order infinite series based on bounded convergent sequences. Additionally, in the case where the infinite sequence converges to a value larger than unity, we categorize the infinite sequence as one which diverges to a higher order of infinity ( than the one where the limit of the sequence is less than unity ). In this case, like the with it the following index ( or a variation of it ), we can associate and associate a convergence exponent with such infinite series. The above index can be used in association with an arbitrary divergent series ( and arrive at its convergence exponent. It enables to provide a unified treatment for ordering arbitrary convergent as well as divergent series. Unlike, may not be a norm). As in Section 2, similar properties of can easily be derived. Details are avoided for brevity. Now, let us consider the case where the unbounded sequence is monotone increasing. In such case, the sequence diverges in (i) linear, (ii) polynomial, (iii) exponential or other manner. The infinite series corresponding to these cases diverge to a higher order of infinity in that order ( i.e (i) corresponds to a lower order of infinity ) As discussed in [Rama2], it is possible to generalize the approach in Section 2, 3 to other classes of infinite series and functions. Efforts are underway to develop a logically consistent set of concepts to order most of the divegent infinite series. Remark 2: In the theory of, various results on of functions as well as infinite dimensional vectors are well documented. These well known results ( as well as ideas ) are invoked to derive new results. For instance Minkowski, Holder inequalities are well known.

8 Suppose we consider two divergent infinite series, whose terms are non negative. Furthermore let for all n. Then, we reason that diverges to a higher order of infinity than the infinite series. More generally, if asymptotically the partial sums of two divergent infinite series can be ordered, then similar inference as above can be made. 5. Conclusions: In this research work, based on the concept of convergence exponent, certain divergent infinite series are ordered. The approach is generalized to non integrable functions. An effort is made to develop a consistent Theory of Orders of Infinity. REFERENCES: [Kno] K. Knopp, Theory and Application of Infinite Series, Dover Publications, Inc., New York [Rama1] G. Rama Murthy, Weakly Short Memory Stochastic Processes: Signal Processing Perspectives, Proceedings of International Conference on Frontiers of Interface between Statistics and Sciences, December 20, 2009 to January 02, 2010, [Rama2] G. Rama Murthy, Orders of Infinity, Manuscript in Preparation