The Basel Problem - Numerous Proofs

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IntroPf 1Pf 2Pf 3Pf 4Pf 5ReferencesThe Basel ProblemNumerous ProofsBrendan W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarApril 11, 2013Brendan W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5ReferencesAbstractTheBasel Problemwas first posed in 1644 and remainedopen for 90 years, until Euler made his first waves in themathematical community by solving it. During his life, hewould present three different solutions to the Problem ,which asks for an evaluation of the infinite series k= then, people have continually looked for new,interesting, and enlightening approaches to this sameproblem. Here, we present 5 different solutions, drawingfrom such diverse areas as complex analysis, calculus,probability, and Hilbert space theory. Along the way, we llgive some indication of the Problem s intrigue W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5References0IntroductionHistoryIntrigue1 Proof:sinxand L H pital2Proof:sinxand MaclaurinAnalysis:sinxas aninfinite product3Proof: Integral on[0,1]24Proof:L2[0,1]and Parseval5Proof: Probability Densities6ReferencesBrendan W.

IntroPf1 Pf2Pf3Pf4Pf5References Abstract TheBasel Problem wasfirstposedin1644andremained openfor90years,untilEulermadehisfirstwavesinthe mathematicalcommunitybysolvingit.

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