[0806.0150] Fun With Fourier Series Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Classical Analysis and ODEs arXiv:0806.0150 (math) [Submitted on 1 Jun 2008 ( v1 ), last revised 27 Apr 2026 (this version, v6)] Title: Fun With Fourier Series Authors: Robert Baillie View a PDF of the paper titled Fun With Fourier Series, by Robert Baillie View PDF HTML (experimental) Abstract: By using computers to do experimental manipulations on Fourier series, we construct additional series with interesting properties. We construct several series whose sums remain unchanged when the $n^{th}$ term is multiplied by $\sin(n)/n$. One example is this classic series for $\pi/4$: \[ \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots = 1 \cdot \frac{\sin(1)}{1} - \frac{1}{3} \cdot \frac{\sin(3)}{3} + \frac{1}{5} \cdot \frac{\sin(5)}{5} - \frac{1}{7} \cdot \frac{\sin(7)}{7} + \dots . \] Another example is \[ \sum_{n=1}^{\infty} \frac{\sin(n)}{n} = \sum_{n=1}^{\infty} \left(\frac{\sin(n)}{n}\right)^2 = \frac{\pi-1}{2}. \] This paper also discusses an included Mathematica package that makes it easy to calculate and graph the Fourier series of many types of functions. Comments: Changes from the June, 2023 version: added Section 12.3 and Equation 12.56. No changes to the code in FS.m Subjects: Classical Analysis and ODEs (math.CA) MSC classes: 40-01, 42-01 Cite as: arXiv:0806.0150 [math.CA] (or arXiv:0806.0150v6 [math.CA] for this version) https://doi.org/10.48550/arXiv.0806.0150 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Robert Baillie [ view email ] [v1] Sun, 1 Jun 2008 14:48:17 UTC (285 KB) [v2] Mon, 29 Jun 2015 15:08:48 UTC (1,128 KB) [v3] Tue, 18 Jul 2017 01:27:47 UTC (1,396 KB) [v4] Tue, 9 May 2023 15:44:45 UTC (756 KB) [v5] Thu, 15 Jun 2023 17:43:52 UTC (757 KB) [v6] Mon, 27 Apr 2026 19:06:44 UTC (758 KB) Full-text links: Access Paper: View a PDF of the paper titled Fun With Fourier Series, by Robert Baillie View PDF HTML (experimental) TeX Source view license Ancillary-file links: Ancillary files ( details ) : FS.m Current browse context: math.CA < prev | next > new | recent | 2008-06 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from