## Splitting Infinite Series into Real and Imaginary Parts

I need a quick reminder that this is (hopefully) true:

Let $\sum a_n$ be an infinite series of complex terms which converges but not absolutely. Then can we still break it up into its real and imaginary parts?

$$\sum a_n = \sum x_n + i\sum y_n$$
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 Quote by Poopsilon I need a quick reminder that this is (hopefully) true: Let $\sum a_n$ be an infinite series of complex terms which converges but not absolutely. Then can we still break it up into its real and imaginary parts? $$\sum a_n = \sum x_n + i\sum y_n$$

Well, since a (complex, real or whatever, as long as we have a definite meaning for infinite sums) series converges iff the sequence of its partial sums converges finitely, and a complex seq. converges iff its real and imaginary parts converge, then...yes.

DonAntonio
 Ok cool, thanks.