8 + 4 + 2 + ...
We can find the sum of an infinite geometric series if and only if the absolute value of the common ratio is smaller than 1.
|r| < 1 &⇔ converges
|r| ≥ 1 &⇔ divergesLet's take a look at the terms of our series!
To get the next term, we multiply by 12. This is the common ratio of the series.
| 1/2|=1/2 < 1
Since the absolute value of the common ratio is less than 1, the series converges, so we can calculate its sum. Recall the formula for the sum of an infinite geometric series.
∑^(∞)_(n=1) ar^(n-1) = a/1-r
In this formula, r is the common ratio and a is the first term in the series. Let's substitute r = 12 and a = 8 to evaluate the sum.
In this case, the common ratio is 3. Let's calculate its absolute value.
| 3| = 3 > 1
The absolute value of the common ratio is greater than 1. This means that the terms in the series increase, so the series diverges. As such, we cannot calculate the sum.
