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Let $(a_n)_n \in \mathbb{R}$ and let $(z_n)_n \in \mathbb{C}$ be two sequences, and let $f_n(x) \in \mathbb{R}$ and $g_n(z) \in \mathbb{C}$ be two sequences of functions.

To check if $\sum a_n \in \mathbb{R}$ converges we can use tests such as the root test, the ratio test, the integral test, the comparison test, etc...

My questions are the following:

1 - Can we apply those tests to the series $\sum z_n \in \mathbb{C}$?

I've learned that we can always write $z_n = x_n + iy_n$ getting that: $\sum z_n = \sum x_n + i \sum y_n$ and we study the convergence of $\sum z_n$ by studying the convergence of $\sum x_n$ and $\sum y_n$. But those 2 series are also series of real valued sequences. My questions is can we apply the tests we apply to these real valued series to the complex valued series directly?

2 - Can we apply those tests to $\sum f_n(x)\in\mathbb{R}$? What about $\sum g_n(z)\in\mathbb{C}$?

Can we apply the same test we use for regular series to series of functions? Because if we fix an $x$ or $z$ this basically became a regular series where $x$ or $z$ is just a constant.

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1 Answer 1

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1 - As long as you don't rely on the order of $\mathbb{R}$. i.e., restrict your discussion to absolute convergence, you're good. You can apply those methods even if your series are formed by elements in a Banach space.

2 - As you said, if you choose some $x$ or $z$ beforehand, you can apply the methods for numerical series. But there are tests tailored specifically to series of functions such as Weiertrass M-test and an analogous one to Dirichlet's theorem, that are helpful to detect uniform convergence, which is beautiful.

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