TITLE: Striking applications of summation by parts
QUESTION [7 upvotes]: 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.

REPLY [9 votes]: 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:
$$\sum_{n=1}^{\infty} \frac{1+\frac12+\frac13+\cdots+\frac1n}{1+2+3+\cdots+n} = \frac{\pi^2}{3} $$