Convergence from fixed-length series blocks ($b_n={\sum }_{j=1}^m{a}_{m(n-1)+j}$)

For a fixed positive integer $m$, if $a_n\to 0$ 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.