Can I change this summation to a sum of other summations? The form of the summation I have is 
$$\sum _{ x=0 }^{ \infty  }{ x{ a }^{ x } } $$
I need to somehow remove the $x$ from the original summation in order to achieve the geometric series in each  other summation. For instance, $$\sum _{ x=? }^{ \infty  }{ { a }^{ x }\quad +\quad  } \sum _{ x=? }^{ \infty  }{ { a }^{ x }\quad +\quad  } \sum _{ x=? }^{ \infty  }{ { a }^{ x }\quad +\quad  } ...$$ I have seen this done before, but forget how to work with the bounds of each new summation . It would be greatly appreciated if anyone had a clue what I was talking about.

This is one of my favourite tricks. Multiplying a series by a carefully chosen term. It is especially useful in dealing with arithmetic-geometric series. 
$$\begin{align}
 S &= a + 2a^2 + 3a^3 + \dots \\
 aS &= a^2 + 2a^3 + 3 a^4 + \dots \\
S - aS &= a + a^2 + a^3 + \dots \\
S &= \frac {a + a^2 + a^3 + \dots}{1-a}\\
S &= \frac{a}{(1-a)^2}\\
\end{align}$$
The last step assumes $\left\lvert a \right\rvert \lt 1$. Otherwise, the infinite series does not converge. 
