the exercise is: Find $f(x)\in\mathbb C[[x]]$ and $g(x)\in\mathbb C[[x^{-1}]]$ with infinitely many non-zero coefficients such that product (in formal calculus, purely algebraically, i.e. summability principle, no convergence here) exists.
Here's what I did so far: let $$f(x)= \sum_{n\geq0}a_nx^n$$ $$g(x)= \sum_{n\leq0}b_nx^n$$and formal product is $$f(x)g(x)=\sum_{n\in \mathbb{Z}}\bigg(\sum_{n_1+n_2=n}a_{n_1}b_{n_2}\bigg)x^n =\sum_{n\in \mathbb{Z}}\bigg(\sum_{i\geq 0}a_ib_{n-i}\bigg)x^n$$ For this to be defined, the second sum needs to be finite of course. So, initially I took $a_i=1$ for even $i$ and 0 otherwise, $b_i=1$ for odd $i$ and 0 otherwise, but this fails already for e.g. $n=1$ in the product. They are intuitively too "dense" so it fails
Then I tried to play with primes, so I let $$f(x)=\sum_{p \text{ prime}}x^p$$ $$g(x)=\sum_{n\leq 0}x^{2n}$$
and I'm pretty sure this product is defined, but I just cannot prove finiteness rigorously for all $n$, can somebody help me? Also, is there a more trivial example which I'm missing?