LEADER 03791nam 22006495 450 001 9910770243803321 005 20240619184032.0 010 $a3-031-37557-2 024 7 $a10.1007/978-3-031-37557-6 035 $a(CKB)29310830300041 035 $a(MiAaPQ)EBC31001781 035 $a(Au-PeEL)EBL31001781 035 $a(DE-He213)978-3-031-37557-6 035 $a(EXLCZ)9929310830300041 100 $a20231207d2023 u| 0 101 0 $aeng 135 $aur||||||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 12$aA Short Book on Long Sums $eInfinite Series for Calculus Students /$fby Fernando Q. Gouvęa 205 $a1st ed. 2023. 210 1$aCham :$cSpringer Nature Switzerland :$cImprint: Springer,$d2023. 215 $a1 online resource (153 pages) 225 1 $aReadings in Mathematics,$x2945-5847 311 08$a9783031375569 327 $a- To the reader -- Getting close with lines -- Getting closer with polynomials -- Going all the way: Convergence -- Power series -- Distant mountains -- Appendix A: SageMath: A (very) short introduction -- Appendix B: Why I do it this way -- Bibliography. 330 $aThis concise textbook introduces calculus students to power series through an informal and captivating narrative that avoids formal proofs but emphasizes understanding the fundamental ideas. Power series?and infinite series in general?are a fundamental tool of pure and applied mathematics. The problems focus on ideas, applications, and creative thinking instead of being repetitive and procedural. Calculus is about functions, so the book turns on two fundamental ideas: using polynomials to approximate a function and representing a function in terms of simpler functions. The derivative is reinterpreted in terms of linear approximations, which then leads to Taylor polynomials and the question of convergence. Enough of the theory of convergence is developed to allow a more complete understanding of power series and their applications. A final chapter looks at the distant horizon and discusses other kinds of series representations. SageMath, a free open-source mathematics software system, is used throughout to do computations, provide examples, and create many graphs. While most problems do not require SageMath, students are encouraged to use it where appropriate. An instructor?s guide with solutions to all the problems is available. The book is intended as a supplementary textbook for calculus courses; lecturers and instructors will find innovative and engaging ways to teach this topic. The informal and conversational tone make the book useful to any student seeking to understand this essential aspect of analysis. 410 0$aReadings in Mathematics,$x2945-5847 606 $aSequences (Mathematics) 606 $aMathematical analysis 606 $aFunctions of real variables 606 $aSuccessions (Matemātica)$2thub 606 $aAnālisi matemātica$2thub 606 $aFuncions de variables reals$2thub 606 $aSequences, Series, Summability 606 $aAnalysis 606 $aReal Functions 608 $aLlibres electrōnics$2thub 615 0$aSequences (Mathematics) 615 0$aMathematical analysis. 615 0$aFunctions of real variables. 615 7$aSuccessions (Matemātica) 615 7$aAnālisi matemātica 615 7$aFuncions de variables reals 615 14$aSequences, Series, Summability. 615 24$aAnalysis. 615 24$aReal Functions. 676 $a515 700 $aGouvęa$b Fernando Q$0382239 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910770243803321 996 $aA Short Book on Long Sums$93660131 997 $aUNINA