Let $X_{1}, X_{2}...$ be i.i.d random variables with distribution $\mathbb{P} (X_{k} \geq x) = \frac{1}{x^a}, x \in \mathbb{N} = \{1,2,3,...\} , a >0$. Let $\mu \in [0,\infty]$ the expected value of $X_{1}$.
For which $a \in \mathbb{R}_+$ is the expected value finite?
Could someone give me a hint for solving this question. Thanks in advance.
Hint:
$$ \sum_{x \geq 1} P(X_1\geq x) = \sum_{x \geq 1} \sum_{i\geq x} P(X_1=i) $$
$$ = \sum_{i \geq 1} \sum_{x =1}^{i} P(X_1=i) = \sum_{i \geq 1} i P(X_1=i) = E[X_1] = \mu$$
(last bullet point in the wiki section 'Basic Properties' here).