The geometric series is known to converge to

MATHEMATICS · Subjective · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

The geometric series ∑n=1∞13n\sum_{n=1}^{\infty}\dfrac{1}{3^{n}} is known to converge to 12\tfrac12. Consider the regrouped series (13+19)+(127+181)+⋯(\tfrac13+\tfrac19)+(\tfrac1{27}+\tfrac1{81})+\cdots. Does this regrouped series converge, and if so, to what value? Give a brief reason for your answer.

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