Question Video: Determining the Convergence of Geometric Series Mathematics

Video Transcript

Is the infinite geometric series 10 add six add 18 over five add 54 over 25 add 162 over 25 and so on convergent?

We know that a series can either converge or diverge. A convergent series is a series which has a finite limit. Otherwise, we have a divergent series. In order for an infinite geometric series to converge, we’re going to need the terms to get smaller and smaller. Otherwise, we find that the terms get bigger and bigger, and the sum of all the terms diverges to ∞. It does appear here that the terms of the sequence are getting smaller, so we’re tempted to say that we’re able to find the sum to be a finite number. But there is a way that we can check.

For an infinite geometric series 𝑎 one add 𝑎 one 𝑟 add 𝑎 one 𝑟 squared and so on where 𝑎 one is the first term and 𝑟 is the common ratio, so this means the number we multiply to get to the next term, as long as this number is between negative one and one, it’s small enough for the terms of the sequence to be getting smaller and smaller and therefore converge. In other words, an infinite geometric series converges as long as the absolute value of 𝑟 is less than one. This just means that 𝑟 is strictly between negative one and one. But what is 𝑟 for this series?

Well, because we multiply each term by 𝑟 to get the next term — for example, we multiply the first term by 𝑟 to get the second term — then if we take the second term and divide it by the first term, we find 𝑟. So for this series, 𝑟 equals the second term, which is six, divided by the first term, which is 10. And in fact, for this, we could use any two consecutive terms, but we’ll use the first and the second term seen as they’re easier numbers to work with. So we have that 𝑟 is equals six over 10 which is, equivalently, three-fifths.

And remember that we said that an infinite geometric series converges if this common ratio 𝑟 is between negative one and one. And because three over five lies between negative one and one, we can say yes; this infinite geometric series does converge.