Need help finding a theorem on divergent real series?

Question

I have read, in a couple of places that I can not now find, that there exists a theorem that says for any divergent real series of infinite length, there exists a convergent complex series, the real series is the projection (stereographic?) of that convergent complex series.  Any help would be appreciated.

Anonymous · Accepted Answer

I'm not familiar with that theorem, although it's clear that if a divergent real series has an unbounded monotone subsequence of partial sums, then those can be stereographically projected from the real line onto the unit circle in the complex plane, forming a complex subsequence on the unit circle that converges to the point (0, i) -- but it's not at all clear such a theorem is true for a divergent real series that does not have an unbounded monotone subsequence of partial sums, for example ∑sin(n)

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Need help finding a theorem on divergent real series?

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