Resumen: We associate some (old) convergent series related to definite integrals with the cyclotomic equation ¿¿ - 1 = 0, for several natural numbers m for example, for ¿=3,¿3 -1=(¿-1)(1+¿+¿2) leads to ¿1¿¿(1/(1+¿+¿2))=¿/(3v3)=(1-1/2)+(1/4 - 1/5) + (1/7 - 1/8) + · · · . In some cases, we express the results in terms of the Dirichlet characters. Generalizations for arbitrary m are well defined but do imply integrals and/or series summations rather involved. Idioma: Inglés DOI: 10.1155/2015/950521 Año: 2015 Publicado en: The Scientific World Journal 2015 (2015), 950521 [7 pp.] ISSN: 2356-6140 Factor impacto SCIMAGO: 0.381 - Environmental Science (miscellaneous) (Q2) - Medicine (miscellaneous) (Q3) - Biochemistry, Genetics and Molecular Biology (miscellaneous) (Q3)