TITLE: Closed form expression of infinite summation
QUESTION [5 upvotes]: What is the idea behind a closed form expression and what is the general way of finding the closed form solution of an infinite summation?
context: closed form solution of $\sum^\infty_{i=1}ia^i$.

REPLY [10 votes]: Lets see:
$\begin{align}
\sum_1^{\infty} ia^i &= a + a^2 + a^2 + a^3 + a^3 + a^3 + ......\\
           &=  \sum_1^{\infty} a^i +  (\sum_2^{\infty} a^i) +  (\sum_3^{\infty} a^i) + ..+  (\sum_{k+1}^{\infty} a^i)+.... \\
          &=  \sum_1^{\infty} a^i + a (\sum_1^{\infty} a^i) + a^2 (\sum_1^{\infty} a^i) + ..+ a^k (\sum_1^{\infty} a^i)+.... \\
           &= \sum_1^{\infty} a^i (1+ a+a^2+....)\\
           &= \sum_1^{\infty} a^i (1+\sum_1^{\infty} a^i)   \\
&= \frac{a}{1-a}(\frac{1}{1-a}) \; \text{if a <1}
\end{align}$ 
The trick is to look for patterns that repeat themselves.