Show that law of large numbers hold

Question

Given,Xi's are iid random variables and

$$f(x)= \frac{1+δ}{x^{2+δ}}$$ $δ>0$ and $X>1$

To show that law of larger numbers hold, I used khinchin's theorem which states that if Xi's are iid then a necessary condition for law of large numbers to hold is that E(Xi) exists. Well, I am unable to show the existence of mean. When I integrate and take limits the answer I am getting is infinity.

Answer 1

We compute \begin{align} \mathbb E[X_1] &= \int_\mathbb R xf_{X_1}(x)\ \mathsf dx\\ &= \int_1^\infty \frac{1+\delta}{x^{1+\delta}}\ \mathsf dx\\ &= \frac{1+\delta}\delta<\infty. \end{align} It follows from the strong law of large numbers that $S_n:=\frac1n\sum_{k=1}^n X_k$ converges almost surely to $\mathbb E[X_1]=\frac{1+\delta}\delta$, i.e. $$\mathbb P\left(\lim_{n\to\infty} S_n = \frac{1+\delta}\delta\right)=1 $$ and from the weak law of large numbers that $S_n$ converges in probability to $\frac{1+\delta}\delta$, i.e. $$\sup_{\varepsilon>0}\lim_{n\to\infty}\mathbb P\left( \left|S_n - \frac{1+\delta}\delta\right|>\varepsilon\right)=0. $$