Closed-form term for $\sum_{i=1}^{\infty} i q^i$ I am interested in the following sum
$$\sum_{i=1}^{\infty} i q^i$$
for some $q<1$. 
Is there a closed-form-term for this? If yes, how does one derive this?
I am also interested in 
$\sum_{i=x}^{\infty} i q^i$
for some $x>1$. 

$$\sum_{i=1}^{\infty} i q^i=q\sum_{i=1}^{\infty} i q^{i-1}=q\frac{d}{dq}\sum_{i=1}^{\infty} q^{i}=q\frac{d}{dq}\sum_{i=0}^{\infty} q^{i}=q\frac{d}{dq}\frac{1}{1-q}=\frac{q}{(1-q)^2}$$
$$\sum_{i=x}^{\infty} i q^i=q\sum_{i=x}^{\infty} i q^{i-1}=q\frac{d}{dq}\sum_{i=x}^{\infty} q^{i}=q\frac{d}{dq}\left[\sum_{i=0}^{\infty} q^{i}-\sum_{i=0}^{x-1} q^{i}\right]=$$$$=q\frac{d}{dq}\left[\frac{1}{1-q}-\frac{1-q^{x}}{1-q}\right]=q\frac{d}{dq}\frac{q^{x}}{1-q}=\frac{(1-x)q^{x+1}+xq^{x}}{(1-q)^2}$$
