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People usually don't like the idea of writing formulas that say they should order their number series in a certain way. Instead, most people prefer to write out an algorithm themselves and then order the numbers accordingly. This is logical because it's much more convenient for them to do it that way, but also because there are some very famous problems with the formulas given by other methods. It may be worth pointing out, however, that you can avoid these problems if you follow this formula. The remainder of this article goes through each case study in detail and provides a detailed explanation of how to avoid any major errors while still producing useful results yourself. A.S. The first case study is for the AS formula. This seems like the simplest of them all, but if you think about it for a second, it becomes clear that this only works on positive numbers and on whole numbers. That means -3 isn't an option here, because it's not a whole number. But there's no problem with 2/8 = 0.25, so let's look at how the coefficient series would look like if we were to use this method: Now let us consider any of these numbers with an odd prime factor, say 11. In this case, our result would be an even number. So there is no reason to use the series at all. Instead, we can just write out a simple formula for our numbers: The tricky part here is to simply avoid using the numbers 11 and 13. For instance, if we included 2/11 = 0.18 this would work, but then 1/11 = 0.09 wouldn't work anymore because it's not a whole number either. That would mean that if we wanted to plug in 3/11 = 0.27, we wouldn't have a series anymore because 3 doesn't fit into the equation correctly [but 1 does]. Ditto for any other number we wanted to use, because our list of possible values would be: 2/11, 1/11, 3/11 and 1/11. The formula we used above for this case is called Horner's Method. It has been around since the time of Newton and Gauss and has been used in all sorts of different ways over the years. The reason it doesn't work correctly is that we can't take into account that many numbers don't fit into it at once. So the problem here is that our list lacks one essential element: an even number which can be divided by 2 without producing an odd remainder. To keep our list of possible values as short as possible, we can just use this simple rule: If x has an even prime factor and is divisible by 2, it does not belong in our list. Here's a table of those numbers:B.S., C.S., D.S., E.S., F.S., G.S., H.S, I. cfa1e77820