\(\Large Y \equiv \displaystyle \sum_{ n = 0 }^{ \infty } (n+1) x^n \)
\(\Large \begin{eqnarray} Y &=& 1+2x+3x^2+ \cdots \\
xY &=& x+2x^2+3x^3 + \cdots \\
\end{eqnarray} \)
\(\Large \begin{eqnarray} Y - xY = (1-x)Y &=& 1+x+2x^2+ \cdots \\
x(1-x)Y &=& x+x^2+x^3 + \cdots \\
\end{eqnarray} \)
\(\Large (1-x)Y-x(1-x)Y = (1-x)^2 Y = 1 \)
\(\Large Y = \frac{1}{(1-x)^2} \)