#1020 · Algebra · Sequences & series

Relation between geometric series sums

For $∣ r ∣ < 1$ , let $S_{p} = 1 + r^{p} + r^{2 p} + \dots$ and $s_{p} = 1 - r^{p} + r^{2 p} - \dots$ denote the sums of the infinite geometric series. Prove that $S_{p} + s_{p} = 2 S_{2 p}$ .

~6 minProof · checked by AIGeometric seriesSeries manipulation

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. V. Geometrical Progression, Examples V.a, No. 26

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