What is the series $\sum_{i=n}^{\infty}\frac{i}{2^i}$ When computing the expected value for a random variable I reached the following series:
$$\sum_{i=n}^{\infty}\frac{i}{2^i}$$
I am confident it is convergent, but have no idea how to compute it.

We can sum your series by interpreting it as the expectation of a random variable, and then computing that expectation in another way.
Suppose a fair coin is tossed repeatedly until it comes up heads. Let $X$ be the number of tosses, up to and including the first head. Let $I=X[X\ge n]$, i.e., $I=X$ if $X\ge n$, and $I=0$ otherwise. It's easy to see that$$E[I]=\sum_{i=n}^{\infty}\frac i{2^i}$$which is the series you asked about.
Let $Y_i$ be the indicator variable whose value is $1$ if $X\ge i$ and $0$ otherwise. Clearly $E[Y_i]=\dfrac1{2^{i-1}}$, the probability of no heads in the first $i-1$ tosses. Moreover, it's easy to see that$$I=nY_n+\sum_{i=n+1}^{\infty}Y_i,$$whence$$E[I]=nE[Y_n]+\sum_{i=n+1}^{\infty}E[Y_i]=\frac n{2^{n-1}}+\sum_{i=n+1}^{\infty}\frac1{2^{i-1}}=\frac n{2^{n-1}}+\frac1{2^{n-1}}=\frac{n+1}{2^{n-1}}.$$
Edit. $n$ is a fixed natural number; $X$ is a certain discrete random variable whose values are nonnegative integers. 
For each $i\in\mathbb N$, the random variable $Y_i$ is a function of $X$, namely, $Y_i$ takes the value $0$ when $X\lt i$, the value $1$ when $X\ge i$. In terms of the discrete form of the Heaviside step function, $Y_i=H[X-i]$; in the Iverson bracket notation, $Y_i=[X\ge i]$.
We also defined a random variable $I=XY_n$; when $X\lt n$ we have $I=X\cdot0=0$, when $X\ge n$ we have $I=X\cdot1=X$.
Now we verify the identity$$I=nY_n+\sum_{i=n+1}^{\infty}Y_i.$$Let $k$ be the observed value of the random variable $X$. By our definition of $Y_i$, we have $Y_i=1$ for all $i\le k$, and $Y_i=0$ for all $i\gt k$.
If $k\lt n$, then all terms on both sides of the identity are zero.
If $k=n$, then the identity reduces to $$n=n\cdot1+\sum_{i=n+1}^{\infty}0.$$
If $k=n+r$ for some $r\in\mathbb N$, then the identity reduces to $$n+r=n\cdot1+\sum_{i=n+1}^{n+r}1+\sum_{i=n+r+1}^{\infty}0.$$
