= Convergence from fixed-length series blocks
{title2=$b_n=\sum_{j=1}^{m}a_{m(n-1)+j}$}
For a fixed positive <integer> $m$, if $a_n\to0$ and the <series> of the displayed block sums converges, then the original <series> converges to the same <limit of a sequence>. Its <partial sum> at a block endpoint is exactly a <partial sum> of $\sum b_n$; the difference at an intermediate endpoint has at most $m-1$ terms, all tending to zero. This justification is essential when block cancellation proves convergence without <absolute convergence>. It does not apply without further control to blocks whose lengths grow.
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