I am having trouble completely reconciling the central limit theorem in its various forms. Specifically it has to do with notation.
If we have a sequence $X_{1}, X_{2}, \dots$ of i.i.d random variables with common mean $\mu$ and common variance $\sigma^{2}$ and define $S_{n} = \sum_{i = 1}^{n}X_{i}$. The central limit theorem states that
$$\lim_{n \to \infty} \bigg(\frac{S_{n} - n\mu}{\sigma \sqrt{n}}\bigg) \to N(n\mu, n\sigma^{2})$$
As well, if we define $\bar{X_{n}} = \frac{\sum_{i = 1}^{n}X_{i}}{n}$. Then the central limit theorem states:
$$\lim_{n \to \infty} \bigg(\frac{\bar{X_{n}} - \mu}{\sigma/ \sqrt{n}}\bigg) \to N(\mu, \frac{\sigma^{2}}{n})$$
My issue is with understanding exactly how the limit is being applied. So for instance in the summation version of the CLT, is the $\infty$ being applied to both sides ? i.e:
$$S_{\infty} = \sum_{i = 0}^{\infty}X_{i} = X_{1} + X_{2} + \dots$$
Or should it be interpreted as taking an infinite amount of iterations of the random variable $S_{n}$ which is defined in terms of the random variables $X_{i}$ for a fixed amount of $n$ values? But just by the simple properties of limits this would not make sense. As can be seen I'm still a bit confused about this part of the concept.
CLT actually states that this $$ \lim_{n \to \infty}\left( \frac{S_n - n\mu}{\sigma \sqrt{n}}\right) $$ converges in distribution to this $$ N(0,1)$$
And $S_n$ converges to $N(n\mu, n\sigma^2)$, which follows from how mean and variance change after linear transformation.
Maybe you should first revise how the statement looks like in general, and what that convergence is https://en.m.wikipedia.org/wiki/Central_limit_theorem