What do LLNs say?
At the risk of getting too oversimplistic, for independent sequence $\langle X_i\rangle_{i\in\mathbb N},$ they say the partial sum (or average, precisely)
$$\frac{1}{n}(X_1+X_2+\ldots+ X_n) \tag 1$$
converges to common mean $\mu$ in certain sense provided second moment or forth moment be finite and uniformly bounded depending on what convergence one is bothered with.
When $\rm i.i.d.$ is imposed, only the first moment needs to be finite.
What is Wikipedia saying?
It is only saying form like
$$\sum_{i=1}^n X_i -n\times \bar X$$
doesn't make sense for
not only does it not converge toward zero as $n$ increases, but it tends to increase in absolute value as $n$ increases.
Finally as Xi'an noted in the comments, be careful with indexing set and limiting variable.
Further Read:
$\rm [I]$ A First Look at Rigorous Probability Theory, Jeffrey S. Rosenthal, World Scientific Publishing, $2006, $ section $5.3,~5.4,$ pp. $60-64.$