E.10 SUMMATION BY PARTS
Summation by parts is a technique for evaluating a finite sum of sequences that is analogous to integration by parts used for continuous functions.
Lemma E.1. For sequences x[k] and y[k]:
(E.66) 
Proof. This expression is verified by bringing the sums to the left-hand side and combining them to cancel the x[k]y[k] terms:
(E.67) 
The right-hand side follows because all other terms in the sum cancel.
Note that other forms are possible, such as
Defining the difference sequences 
and 
, (E.68) can be written as
(E.69) 
This form resembles integration by parts:
(E.70) 
which has been written using Riemann–Stieltjes integrals (see Appendix D).
Example E.3. Summation by parts can be used to verify the finite sum formula in (E.34). Letting x[k] = k and y[k] = xk
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