Pub Date : 2023-07-21DOI: 10.1080/07468342.2023.2223513
Lawrence J. Smolinsky
It is a counterintuitive idea to calculus students that conditionally convergent series may be rearranged to converge to different sums. Some nice examples can be helpful and fascinating to students. This note describes a family of such rearrangements suitable for calculus and undergraduate analysis students. There is one series for each k ∈ N , with k = 1 the original series. The cases of k = 1 and k = 2 are separately presented, as they are particularly easy to show to a calculus class. The general case uses material from a first calculus class but is more involved. Various versions of the series are known and, for k > 1 , wonderful. For each series (labeled k ∈ N ) the positive terms and the negative terms form two harmonic series. The order of the two series are preserved so it is easy to see they are rearrangements.
{"title":"Rearrangements of a Conditionally Convergent Series Summing to Logarithms of Natural Numbers","authors":"Lawrence J. Smolinsky","doi":"10.1080/07468342.2023.2223513","DOIUrl":"https://doi.org/10.1080/07468342.2023.2223513","url":null,"abstract":"It is a counterintuitive idea to calculus students that conditionally convergent series may be rearranged to converge to different sums. Some nice examples can be helpful and fascinating to students. This note describes a family of such rearrangements suitable for calculus and undergraduate analysis students. There is one series for each k ∈ N , with k = 1 the original series. The cases of k = 1 and k = 2 are separately presented, as they are particularly easy to show to a calculus class. The general case uses material from a first calculus class but is more involved. Various versions of the series are known and, for k > 1 , wonderful. For each series (labeled k ∈ N ) the positive terms and the negative terms form two harmonic series. The order of the two series are preserved so it is easy to see they are rearrangements.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44935231","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-19DOI: 10.1080/07468342.2023.2225396
Jean-Paul Truc
Summary This note gives a general sufficient and necessary condition to obtain the direct sum , where E is a finite dimensional space over the field ( or ) and f is a linear transformation of E.
摘要本文给出了获得直和的一般充要条件，其中E是域（或）上的有限维空间，f是E的线性变换。
{"title":"How to Marry the Kernel and the Image","authors":"Jean-Paul Truc","doi":"10.1080/07468342.2023.2225396","DOIUrl":"https://doi.org/10.1080/07468342.2023.2225396","url":null,"abstract":"Summary This note gives a general sufficient and necessary condition to obtain the direct sum , where E is a finite dimensional space over the field ( or ) and f is a linear transformation of E.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44869710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-19DOI: 10.1080/07468342.2023.2225397
J. P. Bonomo, Montana Ferita
Summary The problem of finding a Fibonacci sequence which contains a specified number and has the smallest second element is investigated.
摘要研究了找到一个包含指定数且具有最小第二元素的斐波那契序列的问题。
{"title":"A Small Fib","authors":"J. P. Bonomo, Montana Ferita","doi":"10.1080/07468342.2023.2225397","DOIUrl":"https://doi.org/10.1080/07468342.2023.2225397","url":null,"abstract":"Summary The problem of finding a Fibonacci sequence which contains a specified number and has the smallest second element is investigated.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48622787","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-19DOI: 10.1080/07468342.2023.2225398
H. Sporn
Summary A technique developed by E. Forringer for generating a tree of all primitive Pythagorean triples is extended in order to generate other such trees. These methods are equivalent to multiplying 3 3 matrices by Pythagorean triples to generate other Pythagorean triples.
{"title":"Forringer’s Method and More Trees of Pythagorean Triples","authors":"H. Sporn","doi":"10.1080/07468342.2023.2225398","DOIUrl":"https://doi.org/10.1080/07468342.2023.2225398","url":null,"abstract":"Summary A technique developed by E. Forringer for generating a tree of all primitive Pythagorean triples is extended in order to generate other such trees. These methods are equivalent to multiplying 3 3 matrices by Pythagorean triples to generate other Pythagorean triples.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48048852","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-19DOI: 10.1080/07468342.2023.2225391
Stijn Dierckx
Summary Originally published as a puzzle in 2005 [3], the Perplexing Polynomial Puzzle indeed is perplexing: any given polynomial with nonnegative integer coefficients can be completely determined by just two evaluations. In this article, an extension is made to polynomials with arbitrary integer coefficients, by considering a simple translation with such that the result is a new polynomial with only nonnegative integer coefficients on which the original solution can be used. A proof is given that this is indeed always possible, and a method is constructed to determine a suitable k to do so.
{"title":"An Extension of the Perplexing Polynomial Puzzle","authors":"Stijn Dierckx","doi":"10.1080/07468342.2023.2225391","DOIUrl":"https://doi.org/10.1080/07468342.2023.2225391","url":null,"abstract":"Summary Originally published as a puzzle in 2005 [3], the Perplexing Polynomial Puzzle indeed is perplexing: any given polynomial with nonnegative integer coefficients can be completely determined by just two evaluations. In this article, an extension is made to polynomials with arbitrary integer coefficients, by considering a simple translation with such that the result is a new polynomial with only nonnegative integer coefficients on which the original solution can be used. A proof is given that this is indeed always possible, and a method is constructed to determine a suitable k to do so.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44720121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-17DOI: 10.1080/07468342.2023.2225384
Günhan Caglayan
{"title":"Proof Without Words: A Relationship of Heptagonal and Nonagonal Numbers","authors":"Günhan Caglayan","doi":"10.1080/07468342.2023.2225384","DOIUrl":"https://doi.org/10.1080/07468342.2023.2225384","url":null,"abstract":"","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49578822","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-17DOI: 10.1080/07468342.2023.2224731
Günhan Caglayan
{"title":"Proof Without Words: A Relationship of Pentagonal and Hexagonal Numbers","authors":"Günhan Caglayan","doi":"10.1080/07468342.2023.2224731","DOIUrl":"https://doi.org/10.1080/07468342.2023.2224731","url":null,"abstract":"","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43138502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-30DOI: 10.1080/07468342.2023.2216126
Adam Hammett, Kevin A. Roper
{"title":"A Sterling Look at Stirling’s Constant","authors":"Adam Hammett, Kevin A. Roper","doi":"10.1080/07468342.2023.2216126","DOIUrl":"https://doi.org/10.1080/07468342.2023.2216126","url":null,"abstract":"","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41426016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-27DOI: 10.1080/07468342.2023.2207435
T. Leise, P. Straffin
Summary Media Highlights are short, approximately half-page, reviews intended to help CMJ readers monitor a broad spectrum of publications, web materials, professional activities, and instructional resources. Readers are encouraged to submit items that will be of interest to colleagues in the mathematical community.
{"title":"Media Highlights","authors":"T. Leise, P. Straffin","doi":"10.1080/07468342.2023.2207435","DOIUrl":"https://doi.org/10.1080/07468342.2023.2207435","url":null,"abstract":"Summary Media Highlights are short, approximately half-page, reviews intended to help CMJ readers monitor a broad spectrum of publications, web materials, professional activities, and instructional resources. Readers are encouraged to submit items that will be of interest to colleagues in the mathematical community.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43828513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：