# F Simplified representation of the sum of a geometric series

Back to chapter 6.1.2: “The adjustment process to equilibrium and the multiplier”

First of all, we write the sum of a geometric series \(y\) as follows:

\[\begin{equation} y = 1 + x + x^2 + x^3 + \dots = 1 + \sum_{n = 1}^\infty x^n \tag{F.1} \end{equation}\]

Then, we multiply both sides of equation (F.1) by \(x\):

\[\begin{equation} y x = x + x^2 + x^3 + x^4 + \dots = \sum_{n=1}^{\infty} x^n \tag{F.2} \end{equation}\]

We then subtract equation (F.2) from equation (F.1):

\[y - yx = 1\]

Now, we can factor out \(y\):

\[\Leftrightarrow y (1 - x) = 1\]

Finally, we rearrange for \(y\):

\[\Leftrightarrow y = \frac{1}{1 - x}\]

We see that:

\[ 1 + \sum_{n = 1}^\infty x^n = \frac{1}{1 - x}\]

Here we have assumed that the absolute value of \(x\) is smaller than one, \(|x| < 1\). This assumption is necessary to have the value of the geometric series “converging”.