The notation $\sum_{k=1}^\infty a_k$ always means $$\lim_{n\rightarrow\infty}\sum_{k=1}^n a_k.$$
What about $\sum_{k=-\infty}^\infty a_k$, such as in the Laurent series? Does it always means $$\lim_{n\rightarrow\infty}\sum_{k=-n}^n a_k,$$ or does the meaning depend on the context?
The notation must be used only when $\dagger$ is satisfied. $$\lim_{N_1 \to \infty} \lim_{N_2 \to \infty} \sum_{k=-N_1}^{k=N_2} a_k \text{ and }\lim_{N_2 \to \infty} \lim_{N_1 \to \infty} \sum_{k=-N_1}^{k=N_2} a_k \text{ exists and }$$ $$\lim_{N_1 \to \infty} \lim_{N_2 \to \infty} \sum_{k=-N_1}^{k=N_2} a_k = \lim_{N_2 \to \infty} \lim_{N_1 \to \infty} \sum_{k=-N_1}^{k=N_2} a_k \tag{$\dagger$}$$ If $\dagger$ is not satisfied, the notation doesn't make sense. Sometimes, it can also be used if both $$\lim_{N_1 \to \infty} \lim_{N_2 \to \infty} \sum_{k=-N_1}^{k=N_2} a_k \text{ and }\lim_{N_2 \to \infty} \lim_{N_1 \to \infty} \sum_{k=-N_1}^{k=N_2} a_k$$ are either $+ \infty$ or $- \infty$.