Geometric Series (Math Song)
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A song about the convergence of the geometric series. DorFuchs on Facebook: / dorfuchs DorFuchs on YouTube: / dorfuchs DorFuchs on Twitter: / dorfuchs DorFuchs T-shirts: http://www.dorfuchs.de/t-shirts/ http://www.DorFuchs.de/ more math songs: • Math songs ... and for even more math songs just subscribe. Lyrics: Take one plus a half plus a quarter plus an eighth plus a sixteenth. If you keep doing it you quickly notice that it never ends and that's exactly the stupid thing: How do I know what the result will be for an infinite sum? Well, maybe it will help if we visualize it and then you could try it with a cake. If you take one and a half and a quarter and an eighth, then when you do that you quickly see: You never reach 2, but you get closer and closer, which is why you can simply take 2 as the limit and then say that this series converges to 2 and therefore the infinite sum is exactly 2 and look: the summands are powers of one half. If you simply take an x instead of one half, then you can see that x to the power of i represents the summands, where the sum i counts up from 0 to infinity and this thing is now called the geometric series and pay attention to what I am showing you now, because now it is about whether this series converges and it is guaranteed: The geometric series converges to 1 divided by 1 minus x, at least if the absolute value of x is less than 1, otherwise nothing converges. Before we see what the sum converges to with infinity, let's look at what happens if you only add up to n and then multiply this sum by 1 minus x, because if you multiply out the total, you get this thing minus x times that thing, and now let's take a closer look at the back here. Here x is written with exponents from 0 up to n, and I can see from the prefactor x: The exponent increases by 1 each time. If you simply write it with i starting from 1, it becomes clear that you are subtracting almost the same sum again, so you lose everything except this, and now you divide by 1 minus x, and from here it goes relatively quickly, because for an absolute value of x less than 1 there is a sequence of zeros and the following therefore applies: The geometric series converges to 1 over 1 minus x, at least if the absolute value of x is less than 1, otherwise nothing converges. If the absolute value of x is greater than or equal to 1, then you can see for the sequence x to the power of i that it does not converge to 0 for i to infinity, which is why this series does not converge. Chords with capo +1: Em GA (Prechorus Am7 Bm7 Cmaj7 B7) (End CGD Em (3x) C B7 Em) This video is under a free CC-BY-SA 4.0 license (see https://creativecommons.org/licenses/....
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