Finding the expected value and variance for $Y_n^{(c)}$

Question

I am to find all $\beta > 0$ such that the following series converges: $$\sum \limits_{n = 1}^{\infty} n^{- \beta} \big(X_n - E(X_n) \big). \tag{1}$$ $X_n$ is a random variable with exponential distribution with $\alpha$ parameter $(\alpha > 0)$.

Let's define: $$Y_n = n^{- \beta} \big(X_n - E(X_n) \big).$$ It's easy to calculate that:

1. $E(Y_n) = 0,$
2. $Var(Y_n) = \frac{n^{-2 \beta}}{\alpha^2}.$

From Kolmogorov's two series theorem we can easily justify that for $\beta > \frac{1}{2}$ $(1)$ converges.

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Now I would like to use Kolmogorov's three series theorem to check whether $(1)$ converges for $\beta \le \frac{1}{2}$.

Let's consider: $$Y_n^{(c)} = \begin{cases} Y_n, \text{ for } |Y_n| \le c\\0, \text{ else}\end{cases}$$ My question is how can I find:

1. $E(Y_n^{(c)})$,
2. $Var(Y_n^{(c)})$?