Yes, there on that right side of the equation, is the series we started off with. Look familiar? In case you missed it, thats A.
If I simplify the right side of the equation, I get something very peculiar: So far so good? Now here is where the wizardry happens. I start with a series, A, which is equal to 1–1+1–1+1–1 repeated an infinite number of times. This is where the real magic happens, in fact the other two proofs aren’t possible without this. It allows me to use some of the regular properties of mathematics like commutativity in my equations (which is an axiom I use throughout the article).ĭon’t believe me? Keep reading to find out how I prove this, by proving two equally crazy claims:įirst off, the bread and butter. I also want to say that throughout this article I deal with the concept of countable infinity, a different type of infinity that deals with a infinite set of numbers, but one where if given enough time you could count to any number in the set. “The Cesàro sum is defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the first n partial sums of the series” - Wikipedia. For anyone interested in the mathematics, Cesàro summations assign values to some infinite sums that do not converge in the usual sense. This is because all the series I deal with naturally do not tend to a specific number, so we talk about a different type of sums, namely Cesàro Summations. How could adding positive numbers equal not only a negative, but a negative fraction? What the frac?īefore I begin: It has been pointed out to me that when I talk about sum’s in this article, it is not in the traditional sense of the word. This is what my mom said to me when I told her about this little mathematical anomaly. “What on earth are you talking about? There’s no way that’s true!” - My mom