Splitting Infinite Series into Real and Imaginary Parts

[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?

$$\sum a_n = \sum x_n + i\sum y_n$$

 

[DonAntonio]

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

 

[Poopsilon]

Ok cool, thanks.