Notation of an infinite union

Question

Is there any difference between:

$$ \bigcup_{n =1}^\infty a_{n} \\ \bigcup_{n \in \mathbb{N}} a_{n} $$

From my understanding they both define an infinite union. Is this correct?

Answer 1

If you define $\Bbb N = \{1,2,3,\dots\}$, then yes: the two sets you've defined are identical, and describe the same infinite union.

Note that some define $\Bbb N = \{0,1,2,3,\dots\}$

Answer 2

The only difference it that it sometimes unclear if you consider $0$ as an element of $\mathbb{N}$ so that $$ \mathbb{N}=\{0,1,2,...\} $$

or that $$ \mathbb{N}=\{1,2,...\} $$

The first notation removes this ambiguity and makes things more clear.

At the end - I would say that its a matter of preference and convention, I have seen both used many times interchangeably