# Absolutely convergent series of complex functions.

I have to do the following exercise:

To let $${f_n (z) } _ {n in mathbb {N}}$$ a sequence of complex functions and let $$sum_ {n = 1} ^ infty f_n (z)$$,

Proof of this: if $$sum_ {n = 1} ^ infty | f_n (z) |$$ then converges $$sum_ {n = 1} ^ infty f_n (z)$$ converges.

I know how to prove it for a series $$sum_ {n = 1} ^ infty z_n$$ with complex numbers $$z_n = x_n + iy_n$$ because when $$sum_ {n = 1} ^ infty | z_n |$$ converges, you can watch that $$| x_n | <| z_n |$$ and $$| y_n | <| z_n |$$ then by the comparison criteria the real number series $$sum_ {n = 1} ^ infty | x_n |$$ and $$sum_ {n = 1} ^ infty | y_n |$$ converge and we know for real series that implies that $$sum_ {n = 1} ^ infty x_n$$ and $$sum_ {n = 1} ^ infty y_n$$ converge.

When we call $$R_n = sum_ {k = 1} ^ n x_n$$, $$I_n = sum_ {k = 1} ^ ny_n$$ and $$S_n = sum_ {k = 1} ^ n z_n$$,

And $$lim_ {n rightarrow infty} R_n = x$$, $$lim_ {n rightarrow infty} I_n = y$$, then

$$lim_ {n rightarrow infty} S_n = lim_ {n rightarrow infty} R_n + i lim_ {n rightarrow infty} I_n = x + iy.$$

Then $$S_n$$ converges and $$sum_ {n = 1} ^ infty z_n$$ is also good

Is it enough to call? $${w_n } = {f_n (z) }$$ in my original problem and just apply this proof?