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If I have a finite sequence of $N$ functions $f_n\colon\mathbb{N}\to\mathbb{C}$ and a sequence of complex numbers $z_k$, must it be true that

$$\sum_{n=1}^{N} \sum_{k=1}^\infty f_n(z_k) = \sum_{k=1}^\infty \sum_{n=1}^N f_n(z_k)?$$

It seems that a similar question is addressed at Summation Symbol: Changing the Order, but this question addresses only the case where both sums are finite or both are infinite and doesn't seem to address what happens when we're considering the finite sequence of functions.

Motivation for this question

It seems like an equality of this form is used to prove Lemma 5.4 in the proof of Dirichlet's Theorem on Arithmetic Progressions in http://people.csail.mit.edu/kuat/courses/dirichlet.pdf, but the use of the identity isn't explicit so I'm not sure if I'm understading this right. I think I could make sense of the lemma's proof if the above formula always holds, but I don't know if that is a valid assumption or not.

Any assistance is greatly appreciated!

Math Rules
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1 Answers1

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Without any assumption the statement doesn't hold in general. Let $a_n=1$ and $b_n=-1$ for all $n\in \mathbb{N}$, then we have that $\sum_{n\geqslant 1}(a_n+b_n)=\sum_{n\geqslant 1}0 =0$ but $\sum_{n\geqslant 1}1=\infty $ and $\sum_{n\geqslant 1}(-1)=-\infty $, therefore the sum $\sum_{n\geqslant 1}a_n+\sum_{n\geqslant 1}b_n$ doesn't exists.

Masacroso
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