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Rearrangements of a Conditionally Convergent Series Summing to Logarithms of Natural Numbers 自然数对数和的条件收敛级数的重排
College Mathematics Journal Pub Date : 2023-07-21 DOI: 10.1080/07468342.2023.2223513
Lawrence J. Smolinsky
{"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":" ","pages":""},"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}
引用次数: 0
How to Marry the Kernel and the Image 如何把内核和形象结合起来
College Mathematics Journal Pub Date : 2023-07-19 DOI: 10.1080/07468342.2023.2225396
Jean-Paul Truc
{"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":"54 1","pages":"386 - 387"},"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}
引用次数: 0
A Small Fib 小谎
College Mathematics Journal Pub Date : 2023-07-19 DOI: 10.1080/07468342.2023.2225397
J. P. Bonomo, Montana Ferita
{"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":"54 1","pages":"274 - 289"},"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}
引用次数: 0
Forringer’s Method and More Trees of Pythagorean Triples Forringer方法与毕达哥拉斯三元组的更多树
College Mathematics Journal Pub Date : 2023-07-19 DOI: 10.1080/07468342.2023.2225398
H. Sporn
{"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":"54 1","pages":"269 - 273"},"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}
引用次数: 0
An Extension of the Perplexing Polynomial Puzzle 困惑多项式谜题的一个推广
College Mathematics Journal Pub Date : 2023-07-19 DOI: 10.1080/07468342.2023.2225391
Stijn Dierckx
{"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":"54 1","pages":"299 - 307"},"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}
引用次数: 0
Proof Without Words: A Relationship of Heptagonal and Nonagonal Numbers 无言证明：七进制数与非二进制数的关系
College Mathematics Journal Pub Date : 2023-07-17 DOI: 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":" ","pages":""},"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}
引用次数: 0
Proof Without Words: A Relationship of Pentagonal and Hexagonal Numbers 不用文字的证明:五边形数与六边形数的关系
College Mathematics Journal Pub Date : 2023-07-17 DOI: 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":" ","pages":""},"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}
引用次数: 0
A Sterling Look at Stirling’s Constant 斯特林看斯特林常数
College Mathematics Journal Pub Date : 2023-06-30 DOI: 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":"1 1","pages":""},"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}
引用次数: 0
Media Highlights 媒体亮点
College Mathematics Journal Pub Date : 2023-05-27 DOI: 10.1080/07468342.2023.2207435
T. Leise, P. Straffin
{"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":"54 1","pages":"243 - 250"},"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}
引用次数: 0
About the Cover 关于封面
College Mathematics Journal Pub Date : 2023-05-27 DOI: 10.1080/07468342.2023.2209495
{"title":"About the Cover","authors":"","doi":"10.1080/07468342.2023.2209495","DOIUrl":"https://doi.org/10.1080/07468342.2023.2209495","url":null,"abstract":"","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135950124","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}
引用次数: 0
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