value of summation of $2^i\cdot i$ I'm trying to calculate the value of 
$$2^0\cdot0 + 2^1\cdot1 + 2^2\cdot2 + .... 2^n\cdot n$$
I figured this would be summation $2^i \cdot i$ from $i = 0$ to $n$. But iI'm unable to calculate its value.  
I have tried searching online but haven't been able to find a formula or any property that 
could simplify it (maybe there was which I might not have understood)
P.S.This is not a homework question, I need this value to prove a theorem.

$$\sum^m_{n=0}2^n\times n=2^{m+1}m-2^{m+1}+2$$
The limit as $m\to\infty$ is $\infty$. Or, using the limit test, we directly see that $\sum^\infty_{n=0}2^n\times n$ diverges.
