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surjectivity of sum of subseries of harmonic series

Source: VTRMC 2015 P4

May 5, 2021
series

Problem Statement

Consider the harmonic series ∑n≥11n=1+12+13+…\sum_{n\ge1}\frac1n=1+\frac12+\frac13+\ldots. Prove that every positive rational number can be obtained as an unordered partial sum of this series. (An unordered partial sum may skip some of the terms 1k\frac1k.)
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