Lemma: Summation by Parts
Consider the two sequences
and
.
Let
S N =
be the n-th partial sum. Then for any
0
m
n
we have:
![](../../symbols/{a_n}_n.gif)
![](../../symbols/{b_n}_n.gif)
![](../../symbols/psm{a_n}.gif)
![](../../symbols/le.gif)
![](../../symbols/le.gif)
Proof
The proof is simply a calculation, where the various sums are carefully reindexed:![](../../symbols/qed.gif)