 Dyadic-block alternation of reciprocal terms

For $0\le r<2^n$ and $n\ge 0$, the displayed sequence tends to zero, but its series diverges. The signed sum across the $n$th dyadic block has magnitude ${\sum }_{r=0}^{2^n-1}(2^n+r{)}^{-1}\ge 1/2$. 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.