Can a series be empty?

Question

It sounds contra-intuitive, since it won't exist without elements. However, I am thinking in empty sets, null sequences and empty lists, which indeed exists.

Answer 1

Yes. An empty sum is defined to have the value $0$.

Answer 2

A basic rule about sums is that $\sum\limits_{\alpha\in A}x_\alpha+\sum\limits_{\alpha\in B}x_\alpha=\sum\limits_{\alpha\in A\cup B}x_\alpha$ whenever these sums exist and $A\cap B=\varnothing$. Using this for $B=\varnothing$ and any $A$ imposes that $\sum\limits_{\alpha\in \varnothing}x_\alpha=0$. (Note that, by the same logic, $\prod\limits_{\alpha\in \varnothing}x_\alpha=1$.)