Geometric Series: Clarification and Tools

How do you calculate the sum of an infinite number of things? Normally, that is impossible. However, if the infinite sum is a geometric series, it can be done.

First, I will pick \( 0.35 \), which is a number between \( 0 \) and \( 1 \). The geometric series corresponding to that number is:

$$ 1 + 0.35 + 0.35^2 + 0.35^3 + \cdots $$

Let \( x \) be that infinite series. That is, let \( x = 1 + 0.35 + 0.35^2 + \cdots \). The question is: What is \( x \) equal to? We explain below.

What happens if I multiply \( x \) by \( 0.35 \)? The answer:

\begin{equation} \begin{split} 0.35 x & =0.35 \cdot (1 + 0.35 + 0.35^2 + \cdots) \\ & = 0.35 \cdot 1 + 0.35 \cdot 0.35 + \cdots \\ &= 0.35 + 0.35^2 + 0.35^3 + \cdots \\ &= (1 + 0.35 + 0.35^2 + \cdots) \; – \; 1 \\ &= x \; – \; 1 \end{split} \end{equation}

What is important is that:

$$ 0.35 x = x \, – \, 1 $$

Now, I can solve for x. First, subtract \( 0.35 x \) from both sides of the equation:

$$ 0 = x \, – \, 0.35 x \, – \, 1 $$

Next, add \( 1 \) to both sides of the equation.

$$ 1 = x \, – \, 0.35 x $$

Notice that \( x \; – \; 0.35 x = (1 \; – \; 0.35)x = 0.65 x \). So,

$$ 1 = 0.65 x $$

Dividing both sides of the equation by \( 0.65 \), we see that \( x = 1/0.65 \). So, \( x \) is around \( 1.54 \).

In particular,

$$ 1 + 0.35 + 0.35^2 + 0.35^3 + \cdots \approx 1.54 $$

In general, when calculating such geometric series, use the following formula:

$$ 1 + r + r^2 + r^3 + \cdots = \frac{1}{1 \; – \; r} $$

when \( 0 \leq r < 1 \).