Expected value of a summation

Let $r_n$ be a random variable that returns one of the digits $2, 0, 1, 6$ each with equal probability for all positive integers $n$. Find the value of

$$\mathcal{V}=\mathbb{E} \left[ \sum_{n=1}^{\infty} \frac{r_n}{10^n} \right]$$

where $\mathbb{E}$ denotes the expected value of $x$.

Solution

Since each of $r_n$ are independent and each occurs in only one term, we may interchange the expectation and the summation, thus:

\begin{align*}
\mathbb{E} \left [ \sum_{n=1}^{\infty} \frac{r_n}{10^n} \right ] &=\sum_{n=1}^{\infty} \mathbb{E} \left [ \frac{r_n}{10^n} \right ] \\
 &= \sum_{n=1}^{\infty} \frac{\mathbb{E}\left [ r_n \right ]}{10^n}\\
 &= \sum_{n=1}^{\infty} \frac{\frac{2+0+6+1}{4}}{10^n}\\
 &= \frac{1}{4}
\end{align*}