Infinite Families of Infinite Series With Integer Sums

Summary

Motivated by a desire to show first-year calculus students examples of infinite series whose sums are relatively straightforward to find, we demonstrate a technique to calculate such sums using tools often demonstrated to calculus students. We then use these results, which include an explicit formula for the sums in question, to prove the existence of an infinite family of infinite series whose sums are integers.

Acknowledgment

During the development of this work, Damiano Fulghesu was partially supported by Simons collaboration grant 36031.

Additional information

Funding

During the development of this work, Damiano Fulghesu was partially supported by Simons collaboration grant 36031.

Notes on contributors

Damiano Fulghesu

Damiano Fulghesu ([email protected]) has been a Professor of Mathematics at the Minnesota State University Moorhead since 2010. He received his Doctorate from Scuola Normale Superiore, Pisa, Italy, in 2005. He then had a post-doctoral fellowship at the University of Missouri in Columbia. He is interested in algebra and geometry, the topics of his 12 published papers and over 40 talks given in the United States and Europe. He likes gardening and tending to his aquarium.

James A. Sellers

James A. Sellers ([email protected]) received his Ph.D. from Penn State University in 1992. Over the last 30 years, he has taught at Cedarville University, Penn State University, and the University of Minnesota Duluth. James has published over 100 research papers, and has dedicated much of his career in service to the mathematical community. This includes his service as Secretary of the MAA which he completed in 2022. When he’s not thinking about infinite series or research in number theory, you can often find James playing Candy Crush; as of now, he has completed more than 12,345 levels of the game.

Courtney K. Taylor

Courtney K. Taylor ([email protected]) serves as the Provost of Anderson University. He received his Ph.D. from Purdue University in 2008 for work in algebraic topology. He was chair of the Anderson University Department of Mathematics for the last decade, during which he was involved in a variety of collaborative institutional projects and strategic initiatives. His publications include an open access abstract algebra textbook, and he greatly enjoyed overseeing a multi-year undergraduate research project concerning the geometry of polynomials.