[jason_t]

Here's an example.

Show that sum (-1)^n n/(n^2+1) n = 1 to infinity converges.

Proof: This is an alternating series. Set a_n = n/(n^2+1). Then a_n --> 0.

Set f(x) = x/(x^2+1). Then f'(x) = (1-x^2)/(x^2+1)^2 which is < 0 for x > 1. Therefore f is a decreasing function and this shows that a_n decreases too.

Now apply the alternating series convergence theorem.