Infinite geometric series word problem: repeating decimal

Let's say we have the repeating decimal 0.4008, where the digits 4008 keep on repeating. So if we were to write it out, it would look something like this. 0.400840084008, and it keeps on going forever. What I want you to do right now is pause the video and think about whether you can represent this repeating decimal as an infinite sum, as an infinite series. And then think about whether that infinite series is a geometric series. So I'm assuming you've given a go at it. So let's think about it. So for each term of my infinite series, I'm going to represent one of these repeating patterns of 4008. So, for example, I will make this 4008 my first term. So this could be viewed as 0. And this 4008 represents 0.4008. Then I could make this 4008 my next term, or my next term will represent this 4008. And this 4008 is the same thing as 0.00004008. And then this next 4008, well, that represents 0 point-- and we have eight zeroes-- 1, 2, 3, 4, 5, 6, 7, 8, 4008, 4008. And then we would just keep on going like that forever. So we're just going to keep on going like that forever. So hopefully there's a pattern here. We're essentially throwing four zeroes before the decimal every time. And we can just keep on going like that forever. So this is an infinite sum. It's an infinite series. The next question is, is this a geometric series? Well, in order for it to be a geometric series, to go from one term to the next, you must be multiplying by the same value, by the same common ratio. So what are we multiplying when we go from 0.4008 to this one right over here, where we add four zeros before the 4008? What are we multiplying? Well, we moved the decimal four spots to the left. So we're multiplying by 10 to the negative fourth. Or you could view it as we're multiplying by 0.0001, 10 to the negative 1, 2, 3, 4. To go from here to here, well, same thing. Move the decimal four places to the left. So once again, we're multiplying by 0.0001. And so it looks pretty clear that we have a common ratio of 10 to the negative fourth power. So we can rewrite all of this business as 0.4008 times our common ratio for this first term times our common ratio of 10 to the negative fourth to the 0-th power-- so that gives us that right over there-- plus 0.4008 times 10 to the negative fourth to the first power. And that gives us that value right over there. Plus 0.4008 times 10 to the negative 4 to the second power. And we keep on going. And so in this form, it looks a little bit clearer, like a geometric series, an infinite geometric series. And if we wanted to write that out with sigma notation, we could write this as the sum from k equals 0 to infinity of, well, what's our first term going to be? It's going to be 0.4008 times our common ratio, which we could write out as either 10 to the negative fourth or 0.0001. I'll just write it as 10 to the negative fourth. 10 to the negative fourth to the k-th power, to the k-th power. So the next interesting question-- this clearly can be represented as a geometric series-- is, well, what is the sum? You might say, well, that's just going to be 4008 repeating over and over. But I want to express it as a fraction. And so I want you to pause the video. Use what would you already know about finding the sum of an infinite geometric series to try to express this thing right over here as a fraction. So I'm assuming you've had a go at it. So let's think about it. We've already seen, we've already derived in previous videos, that the sum of an infinite geometric series-- let me do this in a neutral color. If I have a series like this, k equals 0 to infinity of ar to the k power, that this sum is going to be equal to a over 1 minus r. We've derived this actually in several other videos. So in this case, this is going to be-- well, our a here is 0.4008. And it's going to be that over 1 minus our common ratio, minus-- and I'll write it like this-- 0.0001, 1 ten thousandth. So what's this going to be? Well, this is going to be the same thing as 0.4008. If you take 1 minus 1 ten thousandths, or you could do this as 10,000 ten thousandths minus 1 ten thousandth, you're going to have 9,999 ten thousandths. Once again, you could view-- let me write this out just so this doesn't look confusing. 1 is the same thing as 10,000/10,000. And you're subtracting 1/10,000. And so you're going to get 9,999/10,000. And so this is going to be the same thing as 0.4008 times 10,000. So times 10,000 over 9,999. Well, what's this top number times 10,000? Well that's just going to give us 4,008. 4,008 over 9,999. And we've just expressed that repeating decimal as a fraction. So we have succeeded. And you might say, well, maybe we can simplify this thing. And so let's see. This is already a fraction, so we've already kind of achieved it. But if we want to get a little bit simpler. Let's see, if we add the digits up here, 4 plus 8 is 12 and 1 plus 2 is 3. So this up here is divisible by 3. And this down here is clearly divisible by 3. So let's divide both of them by 3. So 3 goes into 4,008-- let's see. It goes into 4 one time, subtract, you get a 10. 3 times 3 is 9. Subtract, you get another 10. Goes into 3 times. 3 times 3 is 9. Subtract. Bring down an 8. 3 into 18 exactly six times. So our numerator is 1,336. This is no longer divisible by 3. The sum of the digits is not divisible by 3, it's not a multiple of 3. And if you divide this bottom number by 3, you get 3,333. And I think we have simplified it. I think we can simplified it about as well as we can. Well, we could check more. Let me know if I didn't. But either way, we have now written this. This was pretty neat. We saw that a repeating decimal can be represented not just as an infinite series, but as an infinite geometric series. And then we were able to use the formula that we derived for the sum of an infinite geometric series to actually express it as a fraction.