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The Relation Between Infinite Series and Improper Integrals by Kermlt

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The Relation Between Infinite Series and Improper Integrals by Kermlt The relation between infinite series and improper integrals Item Type text; Thesis-Reproduction (electronic) Authors Dale, Kermit, 1909- Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 27/09/2021 23:17:22 Link to Item http://hdl.handle.net/10150/551183 The Relation Between Infinite Series and Improper Integrals by Kermlt Dale Submitted In partial fulfillment of the requirements for the degree of : . ,;:v: . ' Master of Solenoe In the Graduate College University of:Arizona 1 9 3 6 Approved: ^ w ^ vCr>« Major professor Date j- .1. * i. £ 9 9 9 / 9 93 5' K 6" ACOOWLSDGrMBHD Grateful acknowledgment is made to Dr* H* B. Leonard at itiioee suggestion and under whose guidance this work has been done and to Dr. A. W. Boldyreff whose sug­ gestions and review of rough drafts of the material hove also done much to facilitate the work. _ . 9938* Table of Contente lUTRODUCTIOI P. 1 PART I — IBZIHITB SBRIBS Seotion l • • . • . • . p. 4 Sequence, aggregate, upper and lower bounds, convergent, divergent, and eseillatory sequences, monotonlo sequences. : ^ Section 2 . • • ....................... p. 11 Infinite series, sum of series, con­ vergence, divergence, and oscillation. Section 3 ........ ...................p.: .14 Criterion of convergence. Section 4 .... • , . , . p. 14 Series of positive terms, convergence, series of importance. Section 5 . • . • . p. 16 Convergence tests— comparison, Canohy1 s test, ratio tests, examples. Section 6 . ...... p. 24 Abel's theorem, integral test— proof and geometric interpretation. Section 7 p. 32 Absolute convergence, conditional oonverginoe, convergence of alternating series. - ' : , ' " * PART II — IMPROPER INTEGRALS Section 1 p. 37 Definition of definite integral. p. 38 Integral over an infinite range, convergence. Table of Contents, continued. Section 5 . • ....................... p. 39 Analogy between improper integrals and infinite series. Section 4. ........................... p. 40 Convergenoe tests. Section 5 • . • . • . p. 45 Absolute convergence, tests for absolute convergence, examples. Section 6 .... , • . p. 48 Analogy with infinite series, strik­ ing property of infinite integrals. Section 7 p. JB1 Integrals expressed as the limit of a sum, examples. Section 6 ....... ............... p. 56 Special methods for testing conver­ gence, Diriehlet integrals, Fresnel integrals. Section 9 ......................... .. p. 60 Integrals in which integrand becomes infinite. Section 1@ . ...........•••• p. 68 Determining convergence of second type improper integral. Section 11 • ............... p. 68 Special methods. IHTReDUCHOI A oomprehensive study of the theory of infinite ser­ ies and improper integrals would he an undertaking quite impossible of being consummated in any short tract such as this and would presuppose a degree of accomplishment and scholarship to which this writer lays no claim. However, a brief study of these topics suffices to impress one with the close similarity between them in method of treatment. A central purpose of this thesis will be to exhibit this similarity in detail as a complete analogy and to draw up­ on wherever possible in order to facilitate the discussion of improper integrals which follows a short study of in­ finite series. The existence of the analogy between infinite series and improper integrals has been noted by such well known mathematicians as Hardy* and Bromwich**, but both seem to assert that the analogy is not complete. Thus, Hardy says, "There is one fundamental property of a convergent infinite series in regard to Shioh the analogy between infinite ser­ ies and infinite integrals breaks down." Bromwich makes a * Hardy, G. H., "A Course of Pare Mathematics." Fourth edition, p. 324. **Bromwloh, T. J., "An Introduction to the Theory of In­ finite Series.” Second edition, p. 469. - 8 - more oarefal statement which is perhaps not shhject to direct challenge bmt is a little misleading in the opinion of this writer. "He says, "But one striking feature pre­ sents itself in the theory of Infinite integrals which has no counterpart in the theory of series.” An attempt will be made in this thesis to show that this "striking feature" constitutes only an apparent and not a real abrogation of the complete analogy. It is not asserted that no property of infinite series can be found which does not hare its analogue in the theory of improper integrals or rice versa, but it is claimed that the proper­ ties of improper integrals discussed in this essay are com­ pletely analogous with those of Infinite series, and this includes the property presented by Hardy as being destruc­ tive of the complete analogy. The scope of the discussion of infinite series will necessarily be limited, being determined largely by what is needed to fit in with the subsequent discussion of im­ proper integrals. Many important topics such as uniform convergence, criteria for term by term differentiation and integration, and the effect of rearranging terms will be omitted completely. Beyond making the discussion of improper integrals coincide as closely as possible with that of Infinite ser­ ies, the aim will be to present the most stiking proper­ ties of these integrals and to investigate by means of - 3. special methods a few important improper integrals which do not yield to general methods. Although borrowing has necessarily been heavy, specific references have been omitted for the most part, since sub­ stantially similar discussions of the borrowed material may be found in a great many sources. A bibliography at the end omtains a list of those to which the writer found it convenient to refer. -4 - 3 ^Part I . IIFIHIIB SMIBS !• The place 0f infinite sequencesand series in the development of modern analysis is a dominant one. These oonoepte and their related studies do not constitute merely an unoonaeoted addition to the store of mathematical know­ ledge; there are haeio, and without them scarcely any of the developments in mathematical analysis of the past two cen­ turies oould have keen carried out. Although the great mathematicians of antiquity accomplished very little in this field, reference to certain special types of series may be found very early in the history of mathematics. .For example, the Greeks studied and summed the geometric series and arithmetic series, hut with analysis scarcely more than incipient it was scarcely to be expected that the great Athenian and Alexandrian geometers would have contributed much toward the clarification of concepts necessary for the understanding of those topics so essential to present day mathematics. A cursory study of mathematical history cannot fail to reveal the prime importance of clearly formed concepts and exactly stated definitions if progress is tc be made in any related branch of mathematics. This is especially true in the study of those topics included under the general olassi- floation of laflnlt# aggregates, since here we are dealing with ideas which are difficult to grasp Intuitively. Sequence. The concept of a sequence Is an abstraction that is formed more or less hazily early in the mind of each individual. He learns to distinguish between a suc­ cession in which events of a set occur or objects of a group pass before his eyes in an indefinite order, and an ordered succession suoh as is met in counting positive integers. Thus, we say that when events of a set occur in some pre­ scribed order we have a sequence. In particular, we speak of a sequence of numbers, ult ug, u3f . , un , . when some rule is given whereby the nth term of the set may be written down. The notation (un) is commonly used to indicate a sequence, i.e., (un) s ult u2, u5, . , • Infinite sequence. If each term of the sequence has a successor the sequence is an infinite sequence. By defini­ tion then, an infinite sequence has no last term, and thus we prepare to erect a superstructure upon a concept which seems to smack of metaphysics. Aggregate. As has been implied, the study of infinite sequences and series may be included under the general head­ ing of infinite aggregates. A sequence of numbers is a particular example of a numerical aggregate• An aggregate may refer to events, objects, points In space or in a plane, complex numbers, real numbers, etc. If we restrict oar- selves to real numbers we say: an aggregate is a system of «*6 «» real numbers or their oorresponding points on a straight line defined in any way vihate#er. Upper and lower bounds. An aggregate may have numbers greater than or less than any number however large or small respectively, in which case the aggregate is unbounded, but if there is a number K such that for every number s of the aggregate s ~ K then the aggregate is said to be bounded above. Similarly^ if there exists a number k such that ■ 6 k for every s, the aggregate is said to be bounded be­ low. Mote that K and k may or may not be upper and lower bounds respectively. Definition. If we indicate an aggregate by the symbol (B), then we say that (B) has an upper bound M if no number of (B) is greater than M and there is a number of (B) great­ er than M - € , no matter how small the arbitrary positive nmaber 6 may be. Similarly, we say that the aggregate (B) has a lower bound m if no number of (B) is less than m and if there is a number of (B) which is less than m -t- € , no matter how small the arbitrary positive number € may be. Returning to our discussion of sequences we notice that each of the following sequences behaves in a different man­ ner as n increases indefinitely: (1) 1, 2, 3, .
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