Can the sum of an infinite geometric series with common ratio 𝑟 be found when the absolute value of 𝑟 is less than one?
Well, first of all, a geometric series is a series with a constant ratio between successive terms, such as the series three add six add 12 add 24 add 48 and so on. The next term is obtained by multiplying the previous term by two. So we call two the common ratio, and we denote this with the letter 𝑟. Notice how the terms of this series are getting bigger and bigger. So if we try to find the sum of this series, we’re going to end up with ∞ because we’re adding larger and larger terms each time.
But if we look at a different infinite geometric series, such as the series one add one over four add one over 16 add one over 64 and so on, the common ratio 𝑟 is one over four. And notice that for this series, the terms are getting smaller and smaller. In fact, they’re approaching zero, so it is going to be possible to find the sum of this infinite geometric series.
So why does this one work? Well, because the common ratio is so small that the next term is smaller than the previous term. And this is only going to happen when the common ratio 𝑟 is bigger than negative one, but smaller than one. Because between those two values, we find that the common ratio is small enough for the terms in the infinite geometric series to get smaller and smaller.
And this is just equivalent to saying that the absolute value of 𝑟 is less than one. So yes, we can find the sum of an infinite geometric series with common ratio 𝑟 when the absolute value of 𝑟 is less than one. And in fact, this is actually the only case when we can find the sum of an infinite geometric series.