# In what type of series can we apply convergence tests?

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.

## 1 Answer

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.