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

\[ \sum_{k = 0}^{\infty} a x^k = \frac{a}{1 - x} \ . \]

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. $\eqref{eq:a}$,

\[ y(1-x) = 1+x+x^2+x^3+\dots \]

where the right-hand side is a geometric series with $a = 1$. Hence

\[ y = \frac{1}{(1-x)^2} \ . \]