Striking applications of summation by parts In the same vein as this question
Striking applications of integration by parts
I'd also like to have a list of some good applications of the discrete version: summation by parts.
 A: Summation by parts is used to prove Abel's Theorem, and is also used in the Cauchy Criterion when testing for convergence.
Additionally, the following paper given here proves to be rather interesting. 
Summation by parts also find many application in recreational mathematics such as IMO questions. 
A: Summation by parts is useful when the summand has a harmonic number $H_n$ in it.  Recall that summation by parts states
$$\sum_{n=1}^m f_n (g_{n+1}-g_n) = f_{m+1} g_{m+1} - f_1 g_1 - \sum_{n=1}^m g_{n+1} (f_{n+1}-f_n) $$
Note that a Harmonic number difference is very simple (e.g., $\frac1{n+1}$).
For example, consider
$$\sum_{n=1}^{\infty} \frac{H_n}{n (n+1)} $$
Note that $g_n = -\frac1n$.  Then the sum is
$$0 - \left ( -\frac{H_1}{1}\right ) - \sum_{n=1}^{\infty} \left [-\frac1{(n+1)^2}\right ] = 1 + \frac{\pi^2}{6} - 1  $$
Thus the sum is
$$\sum_{n=1}^{\infty} \frac{H_n}{n (n+1)} = \frac{\pi^2}{6}$$
which is not trivially derived otherwise.
ADDENDUM
The above result leads to the very interesting-looking summation:
$$\sum_{n=1}^{\infty} \frac{1+\frac12+\frac13+\cdots+\frac1n}{1+2+3+\cdots+n} = \frac{\pi^2}{3} $$
