{"title":"Lehmer反正切和的视觉证明","authors":"Rex H. Wu","doi":"10.1080/0025570x.2023.2266313","DOIUrl":null,"url":null,"abstract":"SummaryWe provide a visual proof to Lehmer’s infinite sum of the arctangents of the inverse of the odd-indexed Fibonacci numbers. A few corollaries follow from the diagram, including Euler’s Machin-like formula and Strassnitzky’s formula.MSC: 11B39 Additional informationNotes on contributorsRex H. WuREX H. WU (MR Author ID: 1293646, ORCID 0000-0003-0970-3741) would like to thank the anonymous reviewer and the Editor for their many generous suggestions. Rex has been working with Garfield’s trapezoid for many years. He recently found more applications of it on the Fibonacci numbers. This article is one of them. Talking about Garfield’s trapezoid, which is named after president James A. Garfield for his proof on the Pythagorean theorem, Rex unexpectedly met his great-great-grandson Mr. Peter Garfield recently.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A Visual Proof of Lehmer’s Arctangent Sum\",\"authors\":\"Rex H. Wu\",\"doi\":\"10.1080/0025570x.2023.2266313\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"SummaryWe provide a visual proof to Lehmer’s infinite sum of the arctangents of the inverse of the odd-indexed Fibonacci numbers. A few corollaries follow from the diagram, including Euler’s Machin-like formula and Strassnitzky’s formula.MSC: 11B39 Additional informationNotes on contributorsRex H. WuREX H. WU (MR Author ID: 1293646, ORCID 0000-0003-0970-3741) would like to thank the anonymous reviewer and the Editor for their many generous suggestions. Rex has been working with Garfield’s trapezoid for many years. He recently found more applications of it on the Fibonacci numbers. This article is one of them. Talking about Garfield’s trapezoid, which is named after president James A. Garfield for his proof on the Pythagorean theorem, Rex unexpectedly met his great-great-grandson Mr. Peter Garfield recently.\",\"PeriodicalId\":18344,\"journal\":{\"name\":\"Mathematics Magazine\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-11-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematics Magazine\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1080/0025570x.2023.2266313\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematics Magazine","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1080/0025570x.2023.2266313","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 0
摘要
摘要给出了奇索引斐波那契数列的倒数的正切无穷和的一个视觉证明。从图中可以得出一些推论，包括欧拉的类机器公式和斯特拉斯尼茨基的公式。作者简介:rex H. WuREX H. WU(作者ID: 1293646, ORCID 0000-0003-0970-3741)在此感谢匿名审稿人和编辑的慷慨建议。雷克斯多年来一直在研究加菲猫的梯形。他最近在斐波那契数列上发现了更多的应用。这篇文章就是其中之一。最近，雷克斯在谈到以证明毕达哥拉斯定理的美国总统詹姆斯·a·加菲尔德命名的加菲猫梯形时，意外地遇到了他的曾曾孙彼得·加菲尔德先生。
SummaryWe provide a visual proof to Lehmer’s infinite sum of the arctangents of the inverse of the odd-indexed Fibonacci numbers. A few corollaries follow from the diagram, including Euler’s Machin-like formula and Strassnitzky’s formula.MSC: 11B39 Additional informationNotes on contributorsRex H. WuREX H. WU (MR Author ID: 1293646, ORCID 0000-0003-0970-3741) would like to thank the anonymous reviewer and the Editor for their many generous suggestions. Rex has been working with Garfield’s trapezoid for many years. He recently found more applications of it on the Fibonacci numbers. This article is one of them. Talking about Garfield’s trapezoid, which is named after president James A. Garfield for his proof on the Pythagorean theorem, Rex unexpectedly met his great-great-grandson Mr. Peter Garfield recently.
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