Finding the solution of a summation The summation is as follows: 
\begin{eqnarray*}
\sum_{i=1}^n\frac{2+i}{2^i}
\end{eqnarray*}
For some reason, I can't seem to figure out what I'm supposed to do here. I tried breaking it up like
\begin{eqnarray*}
 \sum_{i=1}^n\frac{2}{2^i}+\sum_{i=1}^n\frac{i}{2^i} 
\end{eqnarray*}
but I still don't know what I can do with the closed form.
I tried finding a pattern as well by plugging and chugging, and I came up with 
sum from $i = 3$ to $n$
\begin{eqnarray*}
\sum_{i=3}^n\frac{i}{2^i} 
\end{eqnarray*}
but that's just the second half of my first equation.
So confused as to how I might go about solving this one. How would you break this apart to make it easier to work with? 
 A: HINT:
$$x\frac{d}{dx}\sum_{i=1}^nx^{i}=\sum_{i=1}^nix^{i}$$
Now, sum the geometric series and let $x=1/2$.
A: hint
use
$$x+x^2+...x^n=x\frac {1-x^n}{1-x}$$
$$x+2x^2+3x^3+...nx^n=$$
$$x\frac {d}{dx}(x+x^2+....x^n) $$
$$=\frac {x}{(1-x)^2}(nx^{n-1}(x-1)+1-x^n). $$
with $x =\frac {1}{2} $.
A: Breaking it up is a good idea. You have
$$
\sum_{i=1}^n \frac{2}{2^i} + \sum_{i=1}^n \frac{i}{2^i}
$$
With $x = \frac12$, the values of these summations become clearer:
$$
2 \sum_{i=1}^n x^i + \sum_{i=1}^n i x^i
$$
Now we do a little trick: $ix^i$ is $x$ times the derivative of $x^i$. So we get
$$
2 \sum_{i=1}^n x^i + x \frac{d}{dx} \sum_{i=1}^n x^i
$$
Finally, the standard identity for geometric sums gives you
$$
2 \frac{x - x^{n+1}}{1 - x} + x \frac{d}{dx} \left( \frac{x - x^{n+1}}{1 -x}\right),
$$
and you can now take the derivative of that expression, and then plug in $x = \frac12$.
A: First note that
$$
\eqalign{
  & \sum\limits_{1\, \le \,k\, \le \,n} {{k \over {x^{\,k} }}}  = \sum\limits_{1\, \le \,k\, \le \,n} {k\left( {{1 \over x}} \right)} ^{\,k}  = \sum\limits_{1\, \le \,k\, \le \,n} {k\,y^{\,k} }  =   \cr 
  &  = y{d \over {dy}}\sum\limits_{1\, \le \,k\, \le \,n} {y^{\,k} }  = y{d \over {dy}}\left( {y\sum\limits_{0\, \le \,k\, \le \,n - 1} {y^{\,k} } } \right) = y{d \over {dy}}\left( {y{{1 - y^{\,n} } \over {1 - y}}} \right) =   \cr 
  &  = y\left( {{{\left( {1 - y} \right)\left( {1 - \left( {n + 1} \right)y^{\,n} } \right) + \left( {y - y^{\,n + 1} } \right)} \over {\left( {1 - y} \right)^2 }}} \right) =   \cr 
  &  = y\left( {{{1 - \left( {n + 1} \right)y^{\,n}  + n\,y^{\,n + 1} } \over {\left( {1 - y} \right)^2 }}} \right) =   \cr 
  &  = {1 \over x}\left( {{{1 - \left( {n + 1} \right)\left( {1/x} \right)^{\,n}  + n\,\left( {1/x} \right)^{\,n + 1} } \over {\left( {1 - 1/x} \right)^2 }}} \right) =   \cr 
  &  = {{x^{\,n + 1}  - \left( {n + 1} \right)x + n\,} \over {x^{\,n} \left( {x - 1} \right)^2 }} \cr} 
$$
Then it follows easily that
$$
\eqalign{
  & \sum\limits_{1\, \le \,k\, \le \,n} {{{2 + k} \over {2^{\,k} }}}  = \sum\limits_{1\, \le \,k\, \le \,n} {{2 \over {2^{\,k} }}}  + \sum\limits_{1\, \le \,k\, \le \,n} {{k \over {2^{\,k} }}}  =   \cr 
  &  = \sum\limits_{1\, \le \,k\, \le \,n} {\left( {{1 \over 2}} \right)} ^{\,k - 1}  + \sum\limits_{1\, \le \,k\, \le \,n} {{k \over {2^{\,k} }}}  =   \cr 
  &  = \sum\limits_{0\, \le \,k\, \le \,n - 1} {\left( {{1 \over 2}} \right)} ^{\,k}  + \sum\limits_{1\, \le \,k\, \le \,n} {{k \over {2^{\,k} }}}  =   \cr 
  &  = {{1 - \left( {{1 \over 2}} \right)^{\,n} } \over {1 - \left( {{1 \over 2}} \right)}} + {{2^{\,n + 1}  - \left( {n + 1} \right)2 + n\,} \over {2^{\,n} }} =   \cr 
  &  = 2 - {1 \over {2^{\,n - 1} }} + {{2^{\,n + 1}  - 2 - n\,} \over {2^{\,n} }} = {{2^{\,n + 2}  - 4 - n\,} \over {2^{\,n} }} \cr} 
$$
