I'm trying to find two sequences $\{ a_n \}, \{ b_n \}$ such that $\sum a_n$ converges, $\{ b_n \}$ is bounded, $\sum a_n b_n$ converges, but $\sum a_{n+1} b_n$ diverges. If this is not possible, a sketch of the proof would be greatly appreciated.
2026-04-07 17:51:41.1775584301
Series which is a product of sequences which converges, but diverges when one of them is shifted by 1
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Take $$a_{2n}=(-1)^n/n, a_{2n+1}=0, b_{2n}=1, b_{2n+1}=(-1)^n$$