Say if I need to compute the sum of a series like this one:

\begin{equation} y = 1 + 2 x + 3 x^2 + 4 x^3 + \dots \ , \label{eq:a} \end{equation}

where \(|x| < 1\). This series looks like a geometric series in which case the sum can be computed from

\begin{equation*} \sum_{k = 0}^{\infty} a x^k = \frac{a}{1 - x} \ . \end{equation*}

The coefficients vary in Eq. \(\eqref{eq:a}\), so the relation cannot be directly used.

There is a trick to transform a series like above into a geometric series. Multiplying Eq. \(\eqref{eq:a}\) by \(x\),

\begin{equation} xy = x + 2x^2 + 3x^3 + \dots \, \label{eq:b} \end{equation}

Subtracting Eq. \(\eqref{eq:b}\) from Eq. \(\eqref{eq:a}\),

\begin{equation*} y(1-x) = 1+x+x^2+x^3+\dots \end{equation*}

where the right-hand side is a geometric series with \(a = 1\). Hence

\begin{equation*} y = \frac{1}{(1-x)^2} \ . \end{equation*}