Consider the sequence , where z is a complex number of modulus 1. While for z ≠ 1 our sequence is not convergent, it is not hard to see that, on average, it exhibits quite regular behavior. Indeed, using the formula for the sum of a geometric progression, and assuming that z ≠ 1, we have, for any N > M ≥ 0,
which implies that when N - M is large enough, the averages
are small. More formally, we have
This simple ...