## Lemma: Summation by Parts

Consider the two sequences
and
.
Let

*S*be the_{N}=*n*-th partial sum. Then for any*0 m n*we have:### Proof

The proof is simply a calculation, where the various sums are carefully reindexed:

Consider the two sequences
and
.
Let
*S *_{N} =
be the *n*-th partial sum. Then for any
*0 m
n*
we have: