Geometric Series and the 2n rule

The Geometric Series pops up in all sorts of problems. It's so fundamental that the convergence conditions are generally covered in Calculus, even if the proof is above the pay grade of college freshmen. Simply put, if a and r are non-zero real numbers:

Σarⁿ converges to a/(1 - r) if |r| < 1 and diverges otherwise.

Again, this result is eminently useful as a practical limit. However, it also spawns a more theoretical corollary which also comes in handy:

 If {a_n} is a decreasing, non-negative sequence, then Σa_n converges if and only if Σ2ⁿa_2ⁿ converges. There's nothing magic about powers of two, of course; you can use whatever base is most convenient. The point is, you don't have to show that every single term meets the geometric condition. It's enough to show that the sequence is behaving itself on a logarithmic scale. That's often a much easier thing to show.