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”.