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 ...

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