![]() The function F(z) is called the analytic continuation of f(z). So the new function F(z) agrees with the original function f(z) everywhere f(z) is defined, and it's defined at some points outside the domain of f(z). You might figure out a way to construct another function F(z) that is defined in a larger region such that f(z)=F(z) whenever z is in U. We'll call the domain where the function is defined U. Let's say you have a function f(z) that is defined somewhere in the complex plane. One way to tackle the problem is with the idea of analytic continuation in complex analysis. There are meaningful ways to associate the number -1/12 to the series 1+2+3…, but I prefer not to call -1/12 the "sum" of the positive integers. If I said, "I think the limit of this series is some finite number L," I could easily figure out how many terms to add to get as far above the number L as I wanted. It's pretty clear, then, that using the limit definition of convergence for a series, the sum 1+2+3… does not converge. Some, like 1-1+1-1…, might bounce around between different values as we keep adding more terms, and some, like 1+2+3+4. That is the definition of "sum" that mathematicians usually mean when they talk about infinite series, and it basically agrees with our intuitive definition of the words "sum" and "equal."īut not every series is convergent in this sense (we call non-convergent series divergent). Zeno's paradox says that we'll never actually get to 1, but from a limit point of view, we can get as close as we want. After the second term, we're half of the remaining distance to 1, and so on.Ī visual "proof" that 1/2+1/4+1/8.=1. It's pretty easy to see why: after the first term, we're halfway to 1. If L is finite, we call the series convergent. Roughly speaking, we say that the sum of an infinite series is a number L if, as we add more and more terms, we get closer and closer to the number L. The most common way to deal with infinite addition is by using the concept of a limit. We can keep doing this for any finite number of addends (and the laws of arithmetic say that we will get the same answer no matter what order we add them in), but when we try to add an infinite number of terms together, we have to make a choice about what addition means. If you have, for example, three numbers you want to add together, you can add any two of them first and then add the third one to the resulting sum. You put in two numbers, and you get out one number. Showing such a crazy result without qualification only reinforces that view, and in my opinion does a disservice to mathematics."Īddition is a binary operation. Skyskull says, "a depressingly large portion of the population automatically assumes that mathematics is some nonintuitive, bizarre wizardry that only the super-intelligent can possibly fathom. In a post about this video, physicist Dr. Furthermore, the way it is presented contributes to a misconception I often come across as a math educator that mathematicians are arbitrarily changing the rules for no apparent reason, and students have no hope of knowing what is and isn't allowed in a given situation. There is a meaningful way to associate the number -1/12 to the series 1+2+3+4…, but in my opinion, it is misleading to call it the sum of the series. I'm usually a fan of the Numberphile crew, who do a great job making mathematics exciting and accessible, but this video disappointed me. To go the other way, you reverse the addition and subtraction (k+3) mod 12 - 3 will take k to between -3 and 8.A Numberphile video posted earlier this month claims that the sum of all the positive integers is -1/12. For instance, k mod 12 would take k to between 0 and 11, but (k-3) mod 12 + 3 will take k to between 3 and 14. More generally, if you want to shift j numbers from the beginning to the end, you can do (k-j) mod n + j. \color), but by doing the subtraction before doing the modulus, and the addition after, you end up with a different representative. ![]() Let's assume we want to find a formula for the sequence
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