Find the value of $\sum_{p=0}^{\infty}\sum_{q=0}^{\infty} \frac{2^{-p-q}}{1+p+q}$ Find the value of

$$S=\sum_{p=0}^{\infty}\sum_{q=0}^{\infty} \frac{2^{-p-q}}{1+p+q}$$

In the second summation i used change of variable $p+q+1=r$ then we get
$$S=\sum_{p=0}^{\infty}\:\sum_{r=p+1}^{\infty}\frac{2^{1-r}}{r}$$ $\implies$
$$S=2\sum_{p=0}^{\infty}\:\sum_{r=p+1}^{\infty}\frac{1}{r2^r}$$
Any clue here?

An alternative method is to recognise the sum as 
$$S=\int_0^1\sum_{p,q=0}^\infty 2^{-p-q}t^{p+q}\,dt
=\int_0^1\frac{dt}{(1-t/2)^2}$$
etc.
