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Splitting Infinite Series into Real and Imaginary Parts

by Poopsilon
Tags: imaginary, infinite, parts, real, series, splitting
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Poopsilon
#1
Apr8-12, 12:58 PM
P: 294
I need a quick reminder that this is (hopefully) true:

Let [itex]\sum a_n[/itex] 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?

[tex]\sum a_n = \sum x_n + i\sum y_n[/tex]
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DonAntonio
#2
Apr8-12, 01:22 PM
P: 606
Quote Quote by Poopsilon View Post
I need a quick reminder that this is (hopefully) true:

Let [itex]\sum a_n[/itex] 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?

[tex]\sum a_n = \sum x_n + i\sum y_n[/tex]


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
#3
Apr8-12, 01:26 PM
P: 294
Ok cool, thanks.


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