In Rosen's Discrete Mathematics and Its Applications, the connectivity relation of a relation $R$ on a set $A$ is defined to be $$R^*=\bigcup_{n=1}^\infty R^n$$ Because $R^n$ consists of all the pairs $(a,b)$ such that there is a path length of length $n$ from $a\in A$ to $b\in A$, the definition above implies that the connectivity relation contains every possible path from $a$ to $b$, with the only restriction on the length of the path being that it is greater than or equal to one (that is, that a path from $a$ to $b$ actually exists in the relation $R$).
This much I understand; however, Rosen goes on to show that if $A$ contains $n$ elements and there is a path of length at least one in $R$ from $a$ to $b$, then there is such a path from $a$ to $b$ with a length not exceeding $n$. (This lemma makes intuitive sense because a longer path would necessarily form a loop, which could be removed from the path to form a path of length $n$ or less.) Doesn't this mean that any element $(a,b)$ of the relation $R^m$, with $m>n$, must necessarily be a member of $R$, $R^2$, . . . , or $R^n$, and therefore a member of $\bigcup_{i=1}^n R^i$? In other words, if we can specify the cardinality of the set $A$ to be $n$, then can we neglect the terms in the uppermost definition that (in my understanding) do not contribute additional elements to the connectivity relation? Finally, if so, can we say that $$R^*=\bigcup_{i=1}^n R^i$$ which (at least to me) seems like a simpler, more practical definition of the connectivity relation?
If $A$ is a finite set, then you indeed have to look only finitely far. However, if we take an infinite set, we need the first definition. Therefore the first is more general.
If we consider, for example, the neighbour relation $R$ on the natural numbers $\Bbb N$, where $n\mathrel R m$ if and only if $n=m+1$ or $m=n+1$, then $\langle\Bbb N,R\rangle$ is path-connected, but we can find elements for which the shortest path is arbitrarily long.
Another reason to prefer $\bigcup_{i=1}^\infty R^i$ is that it does not need any information about $A$ to be defined. So it's a little "cleaner" perhaps.