# Math Help - Series

1. ## Series

Let $S = 1 + 2x + 3x^{2} + ... + nx^{n-1}$. By considering S - xS, show that

$S = \frac{1 - x^{n}}{(1-x)^{2}} - \frac{nx^{n}}{1-x}$.

Hence find the sum of the first n terms of the series:

1) $\frac{1}{2} + \frac{2}{4} + \frac {3}{8} + \frac{4}{16} + ...$,

2) 1 + 11 + 111 + 1111 + ...

I can prove that $S = \frac{1 - x^{n}}{(1-x)^{2}} - \frac{nx^{n}}{1-x}$ but I am not sure how to use the result to work out the sum of the following series. Thanks for any help

2. Here is a hint on #1.
$\begin{array}{l}
S = 1 + 2x + 3x^2 + \cdots + nx^{n - 1} \\
xS = x + 2x^2 + 3x^3 + \cdots + nx^n \\
x = \frac{1}{2} \Rightarrow \frac{1}{2} + \frac{2}{{2^2 }} + \frac{3}{{2^3 }} + \cdots + \frac{n}{{2^n }} = ? \\
\end{array}$