Find the limiting distribution of $\sum_{i=1}^n \frac{Z_i + 1/n}{\sqrt n}$

Question

this problem does not seem clear to me its solution in this link you can find it What is the limiting distribution of $\sum_{i=1}^{n}(Z_{i}+1/n)/\sqrt{n}$?

I'm learning a little statistics, and I saw this interesting problem:

Suppose that $Z∼N(0, 1)$ and that $Z_1, Z_2, . . .$ are independent. Use moment generating functions to find the limiting distribution of $\sum_{i=1}^n \frac{Z_i + 1/n}{\sqrt n}$

Ok this is the moment generating functions

I don't know how to use this function to find the distribution this theorem can help

Answer 1

1. The random variable

$$W_i=\frac{Z_i+\frac{1}{n}}{\sqrt{n}}\sim N\Big[\frac{1}{n\sqrt{n}};\frac{1}{n}\Big]$$

(this can be easily proved with fundamental transformation theorem)

2. using independence

$$MGF_{\Sigma W}(t)=[MGF_{W}(t)]^n=[e^{\frac{t}{n\sqrt{n}}+\frac{t^2}{2n}}]^n=e^{\frac{t}{\sqrt{n}}+\frac{t^2}{2}}$$

which is the MGF of a gaussian $N(\frac{1}{\sqrt{n}};1)$

Thus the limit distibution when $n\rightarrow {\infty}$ is still a Standard Gaussian