To find the sum of an infinite geometric series, you first need to check if the series converges, which means it has a limit.

An infinite geometric series looks like this:

$S = a + ar + ar^2 + ar^3 + ...$

In this formula:

• $a$ is the first term of the series.
• $r$ is the common ratio. This is the number you multiply each term by to get the next term.

For the series to converge, the absolute value of the common ratio must be less than 1. This means:

$|r| < 1$

If this is true, you can find the total sum using this formula:

$S = \frac{a}{1 - r}$

Let’s look at an example:

Imagine you have this series:

$2 + 1 + 0.5 + 0.25 + ...$

In this case:

• $a = 2$ (the first term),
• $r = 0.5$ (the common ratio).

First, we check if the series converges by looking at the common ratio:

$|0.5| < 1$

Since that’s true, we can use the formula.

Now plug in the numbers:

$S = \frac{2}{1 - 0.5} = \frac{2}{0.5} = 4$

So, the total sum of this infinite series is 4!