Home/Algebra/Sequences, finite series, and recurrences/Infinite geometric series14. Sequences, finite series, and recurrencesInfinite geometric seriesLevel: intermedioIn 14. Sequences, finite series, and recurrencesFormulaCopy LaTeX∑k=0∞ark=a1−r,∣r∣<1\sum_{k=0}^{\infty}ar^k=\frac a{1-r},\quad |r|<1k=0∑∞ark=1−ra,∣r∣<1Tagsalgebratest convergencealgebra formulaLimitsequencessequences series finitasDetailSum of a convergent geometric series.Explore the ideaFound an error? Contact us.Related formulasRelatedFinite geometric sumSn=a11−rn1−r (r≠1)S_n=a_1\frac{1-r^n}{1-r}\;(r\ne1)Sn=a11−r1−rn(r=1)Open formulaOpen in section