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sequence, convergence of alternating sum

Source: VTRMC 2006 P5

June 2, 2021
SequencesConvergenceSummationlimits

Problem Statement

Let {an}\{a_n\} be a monotonically decreasing sequence of positive real numbers with limit 00. Let {bn}\{b_n\} be a rearrangement of the sequence such that for every non-negative integer mm, the terms b3m+1b_{3m+1}, b3m+2b_{3m+2}, b3m+3b_{3m+3} are a rearrangement of the terms a3m+1a_{3m+1}, a3m+2a_{3m+2}, a3m+3a_{3m+3}. Prove or give a counterexample to the following statement: the series ∑n=1∞(−1)nbn\sum_{n=1}^\infty(-1)^nb_n is convergent.
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