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


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.