Law of large numbers: other formulae Let $X_1, X_2, \ldots $ be an infinite sequence of i.i.d. random variables with $E(X_i)=\mu$ and $\mbox{Var}(X_i) < \infty$.
The law of large numbers states $\lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{X_i}{n} = \mu$.
Wikipedia mentions that "other formulas that look similar are not verified".
In other words $\lim_{n \rightarrow \infty} \sum_{i=1}^{n} X_i = n\mu$ is not correct?
Is there a counter-example or an obvious reason why this is not the case?

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. $60-64.$
