= Dyadic-block alternation of reciprocal terms
{title2=$a_{2^n+r}=(-1)^n/(2^n+r)$}
For $0\le r<2^n$ and $n\ge0$, the displayed <sequence> tends to zero, but its <series> diverges. The signed sum across the $n$th dyadic block has magnitude $\sum_{r=0}^{2^n-1}(2^n+r)^{-1}\ge1/2$. Differences of arbitrarily late <partial sums> therefore fail the <Cauchy sequence> condition on <partial sums>. Alternating signs between blocks do not rescue convergence when the total contribution of an entire block stays away from zero.
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